only buy pre-built if you can get a maddening coupon on it, here in the USA, manufacturers occasionally put out some 20-30% coupons on new and refurbished computers, that is about the only time they are worth it over a self-built computer.
but for 450 pounds:
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: Intel Pentium G3420 3.2GHz Dual-Core Processor (£47.56 @ Amazon UK)
Motherboard: Asus H81M-A Micro ATX LGA1150 Motherboard (£40.98 @ Ebuyer)
Memory: Corsair XMS3 8GB (1 x 8GB) DDR3-1333 Memory (£49.67 @ Amazon UK)
Storage: Seagate Pipeline HD 500GB 3.5" 5900RPM Internal Hard Drive (£27.00 @ Amazon UK)
Storage: Western Digital Caviar Blue 160GB 3.5" 7200RPM Internal Hard Drive (£13.99 @ Amazon UK)
Video Card: MSI Radeon R7 260X 2GB Video Card (£103.97 @ Dabs)
Case: Fractal Design Core 1000 MicroATX Mini Tower Case (£26.98 @ Ebuyer)
Power Supply: XFX ProSeries 450W 80+ Bronze Certified ATX Power Supply (£34.14 @ Scan.co.uk)
Optical Drive: Lite-On iHDS118-04 DVD/CD Drive (£10.78 @ Scan.co.uk)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) (£72.59 @ Ebuyer)
Total: £427.66
this is what I could come up with, has a bit of a headroom too.
Pentium G3420, no this isn't the Pentium 3/4 processors from 2000, but rather a Haswell Pentium! It's based off of Intel's latest architecture, and is basically on par with i3's but with no hyper-threading, that's no matter though since CoD (depends on which CoD though I guess) and Minecraft are more single thread applications, so the Pentium's fairly strong single cores will do the job.
2 hard drives, one for normal usage, one for recording, be sure to set your videos to record to a designated drive, and not the one you install windows and your games on.
Radeon R7 260x, a 7790 rebadge that's a good mid ranged card, performs higher than the GTX 650 Ti for reference, which can handle all modern games on decent settings in case you were wondering.
450W is plenty since this rig will only draw at most 270W or so from the wall.
I'm pretty happy with it and was surprised I could find this much for only 450 pounds frankly, it will exceed most minimum requirements you're worried about
in fact if you decide to get more into gaming, this rig can handle newer games like the Bioshocks, or the Battlefields just fine too, just on lower but respectable settings.