I'm sure you can buy many rigs from OcUK as well as receive many other build suggestions based on "OcUK components" only.
But in case of low budget I think there's a conflict between love to the shop you'd like to buy all parts with, and a theoretical overall performance of rig built from other sources (and other than new condition). I mean in such case the best (IMO) way is to buy a mix of used (and/or B-Grade) parts together with new ones as best bang for a buck.
New components might be case and power supply, B-Grade might be graphic card (and motherboard, if you like some warranty), used might be motherboard, RAM, hard disk (by a cost of SSD, to maximize gaming performance) and CPU. If you want some serious attitude, buy new SSD, or HDD (or refurbished by manufacturer). Depending of luck @Ebay (rich source of used components) you might get some decent CPU cooler, and save some money instead of buying new one or B-Grade.
Example of such build (some elements moved to another category) might look like:
NEW
Case 41 pounds
https://www.overclockers.co.uk/showproduct.php?prodid=CA-044-ZA&groupid=2362&catid=736
PSU 650W 58 pounds
https://www.overclockers.co.uk/showproduct.php?prodid=CA-037-SF&groupid=701&catid=123&subcat=2463
memo 8GB 40 pounds
https://www.overclockers.co.uk/showproduct.php?prodid=MY-132-CR&groupid=701&catid=8&subcat=1517
SSD 120GB 50 poinds
https://www.overclockers.co.uk/showproduct.php?prodid=HD-190-SA&groupid=701&catid=2104&subcat=1427
HDD - depending of storage space needed, 45 pounds for 1TB, brand is personal choice as everyone has different luck to different brands (badblocks and other faults, lol)
optical BR drive 35 pounds
https://www.overclockers.co.uk/showproduct.php?prodid=CD-083-PO&groupid=701&catid=10&subcat=314
B-GRADE GFX 240 pounds
https://www.overclockers.co.uk/showproduct.php?prodid=BG-834-AS&groupid=595&catid=683
USED
some P67/Z68/Z77 motherboard upto 40 pounds (or B-Grade seen here for 42)
i5-2500K + BeQuiet freezer I see now for 61.30 (!) at Ebay, but cannot give a link, free speech welcome
Total: 610,30
Plus some soundcard and eventually speakers, if you want more crisp sound = 700 pounds budget fulfilled.