I would say that the biggest problem is whether you can expose your fruits and veggies to enough sunlight. Although this might be a more drawn out method, you could achieve this by planting in large pots on a wooden staging to lift the plants higher up off the ground.
I would try tomatoes, cucumbers, French beans and lettuce and see how things progress. The ripening of the tomatoes would be key to your problem.

Thanks Jimmy. I don't think I'd be able to get containers high enough to make any real difference. I have been reading that certain leafy greens don't need direct sun, so I might try some of those.

Would a window box on an upstairs windowsill get any sun? If so you could try one of the "tumbling" tomatoes meant for hanging baskets.

Sun is essential for most fruit and veg. Tomatoes, for example, simply won't ripen properly without it.

I would suggest growing herbs like parsley, basil and mint, all of which are fine in shade, and a variety of leafy salads.

I agree with everybody else. Most fruits and veggies need at least 6 hours of direct, unobstructed sun per day - every day. What about the roof? I heard of people in the UK renting allotments. Is that a possibility? Perhaps Claire can give you more information on this. She lives in the UK and would be a good person for you to contact. Here is her YouTube link. You can write her.

https://www.youtube.com/user/clairesallotment/videos

I also 'garden' in a small back yard - and fairly well enclosed by neighbouring houses - but I do have a sunny corner - at least when there is any noticeable sunshine I have a corner that gets direct sun for about 3-4 hours in the early afternoons June-August when the sun is high enough to clear the rooftops.

If you haven't already done so, it is worth taking the time to do a proper 'survey' of the light in your yard - you may be surprised at how much light you do get. Plot the areas of sunshine over several weeks. Of course, you need a couple of nice sunny days to start off with!

You may also find that if you do have a sunny corner you can use the heat reflected from brickwork to improve your growing conditions.

ps I don't grow veg but remember my grandfather used to grow rhubarb and leeks in a terrace back yard (given that he had them growing up through clay pipes suggests that they don't need a great deal of sunshine)