The PC in the OP looks good, you could easily live with your old card for a while until you could afford something better, I'd recommend a GTX 560, you don't need to go the extra mile for the Ti version since the regular 560 is pretty damn robust and can be overclocked pretty well from what I've heard.
You can get GTX 560's for about £130 nowadays, they aren't that expensive anymore.
Looking at some passmark benchmark scores, the GT520 in that computer is very poor, your HD 5750/5770 vastly outperforms it and should be able to play most of the games you want to play, enough to tide you over until you get a better card.
Well clearly, the best answer is to build a computer.
Okay. I've come to terms with that. My phobia of attempting new things will have to be put behind me.
I doubt you could get a better computer for £550~, even if you built it yourself