Sorry, I don't live in London, so can't really help with the venue specifically, but is there somewhere nearby like a duck pond or something that you could go to and you could say that you are really looking forward to walking there with your baby and so it will be a special place for you in the future? Make up a nice picnic - it really is the gesture that counts.
As for a ring, I have a fairly cheap engagement ring because my husband was still a student when we got engaged - We could afford to replace it, but I wouldn't because it is special. It is his birthstone that I have which is a blue topaz ( or can be citrine)and really inexpensive - I didn't want my birthstone as that is pearl and I didn't fancy that!
Maybe something like this:
http://www.baxtersjewellers.com/acatalog/Birth stone_Rings_Celtic.html