Hmm... no idea! Let's debug and see where it takes us...
Here's the plan:
(And a bit of notation: when I say a transistor is 'ON', I mean base current flowing with the voltage across the base-emitter ~0.6V, and current flowing into the collector. 'OFF' = no collector current, Vbe ~ 0V).
Since the circuit is a bistable, we should be able to flip it between its two states by turning the alternate transistors off. If the relay is ON, then that means Q2 is turned on (which it obviously isn't supposed to be until the button is pushed). All you need to do to turn Q2 off is to
short the base and emitter together (in practice this is easily performed with a pair of needle-nosed tweezers or a flat bladed screwdriver) - this will rob the base of drive current, and force the transistor off. This
should turn the relay off - if it doesn't, check that Q2 and/or the relay is wired correctly.
Turning Q2 off should immediately turn Q1 on. This will be evidenced by the relay clicking off and STAYING off after you short the base-emitter of Q2. If it clicks back on the instant you remove the short, then either Q1 is dead, wired incorrectly or there's something amiss in the circuitry around the base. Check the 10K resistor and that the 1N4001 is the right way around. The voltage on the base of Q1 should measure ~0.6V while you have the Q2 short in place.
If the relay clicks off as it should, pressing the button should turn it on, since the button simply shorts the base-emitter of Q1 in the same way you just manually did Q2. Shorting Q2 should then start the whole cycle over again...
If this all looks good, then the circuit is simply waking up in the wrong state. The brute-force thing to try would be to increase the 10 uF capacitor to ~ 100uF and perhaps even parallel it with a 10K resistor. I could start guessing about reasons for Q1's delayed turn on (is the 100n capacitor in place?) but I'll hold off hypothesising until I hear how you go
Let me know how you fare, as well as any observations along the way which might offer clues...
Good luck!