My CPU+mobo+ram is 4 years old (E8500). My videocard is 3.5 years old (gtx260). I can run Skyrim reasonably fine. (25-60fps, ultra settings, 4xAA, 1920x1200). I think you can buy a PC today that will still be valuable in 3-4 years. 5 Years might be stretching it a little. So your goal should be to buy a system that is reasonably high performance, but not at the highest price. Just a notch or two below the top. That is where you get the most value for your pounds.
Do you have a PC currently ? If so, you can reuse parts. Like the monitor (big expense, say 130 pound). The case (~50 pounds). The PowerSUpply (PSU). And the Operating System (I suppose anything but Windows7 would be a bad choice today). Maybe even the HardDisk (HDD).
The gaming performance (the fps) of your computer will mainly depend on 4 items.
1) The CPU. Get an Intel i5-2500k. 160 pounds.
2) The Videocard. Get a Nvidia gtx560ti (note the ti at the end). (Or a AMD 6950 if you want an AMD/ATI videocard). 190 pounds. But worth the money.
3) Motherboard. Anyone based on the Z68 chipset will do. I like the ASRock brand. Roughly 85 pounds.
4) RAM. 4GB is still enough, and I expect will still be enough for most games in the next 3 years. 35 pounds. Type should be DDR-3-12800.
Prices are based on what I saw in the cheapest/best Dutch webshops. UK webshops should have comparable prices.
Total price: 470 pounds.
That's the price for the core of your system. Not cheap, but really worth the money. You'll get high framerates at high settings in most games. You will need to get other components for your old computer, or buy them anew. That will make the difference. Monitor 130 pounds, case 50, PSU 35, HDD 60+. Maybe mouse and keyboard even. Could add up to another 200-300 pounds. (Monitors are expensive

)
Anyway, I hope this gives you some indication on how much you'll have to spend for a nice gaming PC.