What about good old East Lane on Sunday Mornings - the side street near Portland Street. Not quite SE1 - but close enough.
I have seen tomato plants for sale at the Borough Market - on the corner of Southwark Street and Stoney Street.
I expect youve got enough ideas by now but if not, here's a few community-minded ones. I'm not especially recommending them for particular varieties or even particular vegetables (if you'll excuse me jumping in and not answering the question as asked), just for sources of non-commercial plants in general, and because they are all interesting places to visit if youve got the time.
1. St Mungos hostel, round the back of the Welsh Chapel in Southwark Bridge Rd have a thriving gardening project run by a chap called Martin. They often have plants to sell, best time to call is Wednesday mornings. Pot luck (pun intended) what they have on the go though.
2. Not in SE1 but nearby, and definitely worth a visit in its own right is Surrey Docks City Farm, Rotherhithe St. 020-7231-1010
Easy on the 381 bus.
They also have horticulture projects and plants for sale, again pot luck. And bring home a bagful of manure for your toms and cucumbers while you're at it!
3. Not in SE1 and not even near, but a great day out: Plumstead Make Merry fair. I believe this year its on Sat 8th May. Its a general fayre and great fun, on Plumstead Common. It has several stalls selling great home-produced plants. Best stall is Friends of Plumstead Common.
Good luck Ivanhoe, I'll water yours when you go away if you'll do mine (hoping against hope that you might accept this exchange before finding out exactly how many plants I've got)!
Agree with Rabbie that you should go to Roots and Shoots. Its great for kids so I could lend you a couple. You might even be able to trade them in.
I think the Museum of Garden History, by Lambeth Bridge, has an Open Day/plant sale this weekend. (Sunday) Unlikely to get tomatos but they do have good quality plants. It is a curious place and worth visiting.