Ok, I will try to explain it differently.
The local loop is the connection between the customer and the telco. Usually it is one or two telephone cables. Each telephone cable consists of two copper wires.
ISDN is a fully digital network, as the name implies (Integrated Services Digital Network). There are two "flavors" of ISDN, namely, Basic Rate ISDN (BRI), also called Basic Rate Access (BRA) ISDN, and Primary Rate ISDN (PRI), also called PRA ISDN. The connection between the Line Termination (LT) unit, located at the end switch in the central office of the telco, and the Network Termination (NT) unit at the customer side is referred to as U-interface. The BRI U-frame consists of the following fields: two bearer channels (B-channels), namely B1 and B2, with bit rate 64 kbps each, one delta channel (D-channel) with bit rate 16 kbps, and overhead for framing and synchronization with a bit rate of 16 kbps, for a total bit rate of 64+64+16+16 = 160 kbps. B1 can be used for digital voice and B2 for data (at the same time), or can be combined ("bonded" as we say) to form a single 128 kbps data channel (in that case, we cannot have voice service). The D channel is used for signaling (i.e., control), but if we wish it can be used also to carry low-rate packet-switched data, such as X.25 packets at a rate of 9.6 kbps. (I remind you that ISDN is a circuit-switched technology, that is, a physical end-to-end path or circuit has to be established prior communication takes place - this is called call setup, and it is one of the reasons that we need the D-channel, i.e. to establish, monitor, and terminate these calls; in other words, the ISDN is in fact a dial-up technology, since we have a call; anyway, what I want to point here is that the B-channels can carry circuit-switched voice or data.) BRI signal is often called 2B+D signal (two B channels and one D channel, plus overhead of course). The spectrum of this signal depends on the line coding used. In N. America the line coding is 2B1Q (2 binary - 1 quartenary). In other words, we have a mapping of pairs of bits (or dibits) to quartenary symbols, that is, we have 4 different symbol states on the line. Thus, the corresponding symbol rate (baud rate, signaling rate) for this bit rate of 160 kbps is 160,000/2 (since we have 2 bits/symbol) = 80 kHz. In Europe is used a line coding technique called 4B3T (4 binary - 3 ternary) which leads to a baud rate of 160 * (3/4) = 120 kHz. The U-interface is a single pair (two wires) connection, but still offers bi-directional communication (i.e., full-duplex). How the different signals are separated? Usign a technique called echo cancellation. It is complex to describe it here, there are many resources on the topic available on the Internet. Simply stated, the system measures the channel and locates the position and duration of echoes, and then removes them. This can be done either using a Digital Signal Processor (DSP) or special software.
The second version of ISDN, PRI, is quite different. In N. America, we have 23 B channels and 1 D channel, but this time the D channel has bit rate of 64 kbps, plus 1 bit for frame synchronization with bit rate of 8 kbps, for a total bit rate of 24 * 64 + 8 = 1.544 Mbps. This is a PCM-TDM frame referred to as PCM24. Each channel is a sample represented by 8 bits. Each sample is a timeslot (TS) - D-channel is placed in TS0. The frame repeats 8,000 times per second, thus it has a duration 1/8,000 = 125 μs. Each TS has a bit rate, as we said, 8 bits/TS * 1 TS / frame * 8,000 frames/s = 64 kbps. We have in total 8 bits/TS * 24 TS + 1 bit = 192+1 = 193 bits. This signal is the Level 1 of the North American Plesiochronous Digital Hierarchy (PDH), that is, the T-carrier system, and it is referred to as T1. In Europe, we have 32 TS - 30 B channels, 1 D channel of 64 kbps (TS16), and 1 TS for frame synchronization (TS0). This is the so-caleld PCM30 frame, with bit rate 32 TS * 64 kbps/TS = 2.048 Mbps. The total number of bits is 32 TS * 8 bits/TS = 256 bits. As you may have already realized, again each TS is a sample represented by 8 bits, and the PCM frame repeats with a frequency of 8 kHz, that is, it has a duration of 125 μs. This is the so-called E1 signal, the Level 1 of the European PDH, a.k.a. E-carrier. As you can see, in PRI ISDN the line is time-shared between the different channels. The line coding used in N. America originally was AMI (Alternate Mark Inversion), now it is B8ZS (Bipolar 8 zeroes Substitution); the line coding used in Europe is HDB3 (High Density Bipolar 3 Zeroes). B8ZS and HDB3 use substitution codes to elliminate the precense of more than 8 or more than 3 consecutive zeroes respectively, which could lead to loss of sycnhronization due to the DC level. Besides that, are very similar in that they are both binary codes, that is, we have two states on the line, a binary 0 and a binary 1. Thus, the baud rate reflects the bit rate, that is, it is about 1.5 MHz in the U.S. and 2 MHz in Europe.
I have to say that in BRI there is also one other interface, called S/T interface, which connects the NT to the Terminal Equipment (TE) - such as the ISDN telephone. This is a 4-wire (2 pair) interface, and it is mainly used in Europe, not in the U.S. Actually, the BRI ISDN was never popular in the U.S., only the PRI was. The S/T frame consits of the 2B+D channel and 48 kbps overhead (for monitoring, testing, etc.), for a total of 192 kbps. The channels are time-multiplexed in a complex way that I will not refer here. The line coding is Inverse AMI, also called Alternate Space Inversion (ASI) or pseudo-ternary code. In AMI, pulses alternate between binary 1 (Mark) adn binary 0 (Space) in order to ensure the absence of a DC level. The same happens in ASI, with one difference: a binary 1 is represented by an absence of pulse (Space) and a binary 0 by a pulse (Mark).
Something else that I forgot to mention is that in PRI , there is a Private Branch Exchange (PBX), also referred to as NT2, before the NT (also referred to as NT1) at the customer side. The PBX is a concentrator/multiplexer, or simply stated, a switch which distributes the calls (B-channels of the PRI U-frame) to the corresponding Terminal Equipments or collects the calls of the TEs and puts them to the corresponding TS of the PCM frame to send them towards the telco.
To sum up: ISDN is digital. ISDN is a dial-up technology (oh yes, it is), but much better than the original dial-up modem technology (also called analog modem, even though digital modulation schemes are used!), due to the digital signaling, the digital TDM technology, the higher bandwidth used, and other stuff that have to do with upper layers (because ISDN is actually a protocol suite, and defines protocols in layers 1, 2, and 3 - not only in the Physical layer of the OSI Reference Model). In U-interface, when we have BRI ISDN, full-duplex communication is achieved via echo cancellation, whereas when we have PRI via TDM (time-sharing).
This is by no means a complete description of ISDN. I have left "blank" many topics on purpose.
In ADSL now, we take blocks of N symbols of the data stream (each symbol is a group of bits, from 1 bit to m bits) and we convert them from serial to parallel. Then, each one of these N parallel symbols modulates one of N sub-carriers, each one with different frequency. Finally, we sum all these signals to take the resulting composite signal. As you can see, we do not have multiplexing here, because all these sub-carriers carry data symbols from the same source. In multiplexing signals from many different sources share a single medium (let's say a cable), either in time (Time-Division Multiplexing, TDM: each source signal is assigned one or more timeslots) or in frequency (Frequency Division Multiplexing, FDM: each source signal is assigned one or more frequency bands), or in code (Code Division Multiplexing, CDM: each source signal is assigned a code; thus, all signals share the whole bandwidth for the whole time but still they are distinguishable at the receiver due to their different code, which serves as an ID), etc. In ADSL we have multi-carrier modulation (MCM), reffered as Discrete Multi-Tone (DMT): data from a single source are converted from serial to parallel (S/P conversion) and modulate multiple parallel carriers (or tones, that is why the name "multi-tone"). The frequency band for US and DS is different, but they can overlap also and separated at the receiver via echo cancellation. The name "discrete" stems from the fact that the signal is designed in frequency as frequency samples and converted in the time domain as time samples via a DSP which implements the Inverse Discrete Fourier Transform (IDFT) operation using the Inverse Fast Fourier Transform (IFFT) algorithm, and then via a Digital-to-Analog Converter (DAC or D/A) this digital signal is converted to an analog signal for transmission over the channel. At the receiver we have the reverse process: A/D, DFT, and P/S conversion.
As you can see, at the local loop we do not have multiplexing. Only at the telco side we have, where multiple DSL signals are multiplexed in the DSLAM (DSL Access Multiplexer) in order to be transported more efficiently to the core network. But this principle stands at every telecommunication network.
Why we use DMT in ADSL? Because we want high data rate R. This implies high bandwidth W (remember Shannon: C = Blog2(1 + S/N), that is why in ADSL B> 1 MHz (compare with analog telephony, a.k.a. PSTN where the bandwidth is 4 kHz). This implies also low symbol duration T. Thus, typically T<< Td, the channel impulse response. Thus, we will have inter-symbol interference (ISI), that is, the pulses will spread in time and subsequent pulses will interfere at the receiver (i.e., overlap), causing it possibly to interpret some symbols wrong (simply stated, to think that a 0 is a 1 or vice versa, due to the enrgy spreaded at adjacent symbols) and increasing that way the bit error rate (BER). Instead of using complex and costly channel equalizers (digital filters that remove ISI), in DMT the data stream with rate R is converted to N parallel streams with rate RN = R/N. That is, the total bandwidth B is divided to N sub-channels with bandwidth BN = B/N. In other words, the symbol time becomes TN = NT, which, for N sufficiently large, is greater that Td. Thus, we have no ISI! Moreover, the data rate is equal to the original of a single carrier modulation (SCM) system: we have N sub-channels with rate R/N, thus the total rate is again N * R/N = R. In fact, we can use rate adaptation, also called adaptive modulation and coding (AMC), to further improve the performance of the system. How that works? We use some pilot tones (unmodulated carriers) to measure the channel in US and DS. In channels with good quality, we use more bits to modulate the carrier, that is, we have more symbols (higher data rate), whereas in channels with not so good quality we have lower data rate (in order to ensure higher tolerance to distortion), and really bad sub-channels are not sued at all.
How ADSL coexists with POTS and ISDN? Simply, the ADSL spectrum starts over the spectrum of these systems. For example, for ADSL over POTS, we have POTS from 0 up to 4 kHz, we have a guard band from 4 up to 25 kHz, and ADSL US starts at 25 kHz. For ADSL over ISDN, ISDN goes up to 120 kHz and ADSL starts at 138 kHz. Moreover, filters are used to separate the spectrums (in case that energy passes from one spectrum to another due to inefficient filtering). There are systems though where ADSL spectrum overlaps with POTS and ISDN spectrum (I think this is called "naked DSL", but I am not sure; search about that ...). Echo cancellation is used once again.
I tried to give you a detailed but still easy to follow answer. I cannot cover everything in a forum, and I am not an expert, I am just a student who is still learning and tries to improve himself. So, if you find anything that I have explain it wrong, please correct me. I hope I helped you a little bit.