Price To Performance over Upgradability?

Probably you can get something with a bit more money..

PCPartPicker part list: https://uk.pcpartpicker.com/list/3NFqzM
Price breakdown by merchant: https://uk.pcpartpicker.com/list/3NFqzM/by_merchant/

CPU: Intel - Pentium G4560 3.5GHz Dual-Core Processor (£63.47 @ BT Shop)
Motherboard: ASRock - H110M-HDS Micro ATX LGA1151 Motherboard (£46.97 @ Amazon UK)
Memory: Crucial - 8GB (1 x 8GB) DDR4-2133 Memory (£56.40 @ Aria PC)
Storage: Seagate - BarraCuda 1TB 3.5" 7200RPM Internal Hard Drive (£37.97 @ Amazon UK)
Video Card: PowerColor - Radeon RX 560 2GB Red Dragon OC Video Card (£99.95 @ Overclockers.co.uk)
Case: BitFenix - Comrade ATX Mid Tower Case (£34.99 @ Overclockers.co.uk)
Power Supply: EVGA - 430W 80+ Certified ATX Power Supply (£33.79 @ Aria PC)
Total: £373.54
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-24 17:41 BST+0100

PCPartPicker part list / Price breakdown by merchant

CPU: AMD - Ryzen 5 1400 3.2GHz Quad-Core Processor (£146.96 @ More Computers)
Motherboard: Asus - PRIME B350-PLUS ATX AM4 Motherboard (£75.44 @ Amazon UK)
Memory: Corsair - Vengeance LPX 16GB (2 x 8GB) DDR4-3000 Memory (£125.99 @ Aria PC)
Total: £348.39
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-24 17:39 BST+0100

This gets you some decent DDR4-3000 RAM, the cheapest Ryzen CPU (which is more than sufficient for gaming), & a decent B350 chipset motherboard (which not only will support faster Ryzen 5 & 7 chips, but also lets you overclock).

PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Pentium G4620 3.7GHz Dual-Core Processor (£86.21 @ CCL Computers)
Motherboard: MSI - Z270 GAMING PLUS ATX LGA1151 Motherboard (£89.52 @ Amazon UK)
Memory: Corsair - Vengeance LPX 16GB (2 x 8GB) DDR4-3000 Memory (£125.99 @ Aria PC)
Total: £301.72
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-24 17:42 BST+0100

This is a cheaper build...but while you get a 2C/4T CPU with it, you can't overclock with this CPU -- you need at least the i3-7350K, or an unlocked Core i5/i7 to do that -- & while for most games a 2C/4T CPU will be OK, there are a number of games that simply perform much better on 4C/4T or higher CPUs (& they'll only start taking advantage of more & more cores in the future).

PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Core i3-7350K 4.2GHz Dual-Core Processor (£149.99 @ Overclockers.co.uk)
CPU Cooler: Cooler Master - Hyper TX3 Evo 43.1 CFM CPU Cooler (£19.13 @ Novatech)
Motherboard: MSI - Z270 GAMING PLUS ATX LGA1151 Motherboard (£89.52 @ Amazon UK)
Memory: Corsair - Vengeance LPX 16GB (2 x 8GB) DDR4-3000 Memory (£125.99 @ Aria PC)
Total: £384.63
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-24 17:45 BST+0100

This would be the minimum Intel-based build if you wanted to have overclocking capability from day 1. Note that it's about £36 more expensive than the Ryzen 5-based build, & that a) you have to buy a CPU cooler for it (Intel doesn't provide them with unlocked CPUs), & b) for the most part the Ryzen-based build will match or exceed the i3-7350K's performance. Bumping up to the i5-7600K adds another ~£60 to the cost (https://uk.pcpartpicker.com/list/NpnK9W), putting you almost £100 more than the Ryzen 5 1400 build.

And yes, I didn't include storage, GPU, DVD-drive, OS, etc. Two reasons:
#1: There's always the chance that you already have those with an existing system, so you can simply reuse your existing hardware & reinstall your existing Windows (unless it's the OEM version).
#2: It helps showcase that, for £300, you really cannot do any kind of meaningful gaming system (unless you stick with the Intel build & are content to use only the integrated graphics). Even if you decided to go as bare-bones as possible -- Pentium CPU, mATX motherboard, & only 8GB of DDR4-2133 RAM -- you would still have only maybe £95 to spend on your GPU & still stay within budget (https://uk.pcpartpicker.com/list/X4kmTH). The GPU options in that price range aren't exactly stellar -- we're talking about GT 750Ti/1030 or RX 460/550 (https://uk.pcpartpicker.com/products/video-card/#sort=price&page=1), most of which aren't much better than the integrated graphics to begin with.

What all needs to be included in that budget? Windows, any peripherals?

Best I could do and stay close to your budget:
PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Pentium G4600 3.6GHz Dual-Core Processor (£73.80 @ Alza)
Motherboard: Gigabyte - GA-B250M-DS3H Micro ATX LGA1151 Motherboard (£65.12 @ CCL Computers)
Memory: G.Skill - Ripjaws 4 series 8GB (2 x 4GB) DDR4-2400 Memory (£59.30 @ Amazon UK)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£39.95 @ Aria PC)
Case: Rosewill - SRM-01 MicroATX Mini Tower Case (£26.99 @ Amazon UK)
Power Supply: Corsair - CXM 450W 80+ Bronze Certified Semi-Modular ATX Power Supply (£49.89 @ Ebuyer)
Total: £315.05
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-24 19:14 BST+0100

Not sure how good even light gaming is going to be on integrated graphics, but the G4600 has a little better iGPU than the G4560, for an extra £10. Has decent upgrade potential: can drop in another 2x4 GB RAM and discrete GPU (should be able to run up a GTX 1060 with that PSU) down the road. Can also upgrade the CPU all the way up to a i7-7700(K) with that mobo.

OK, respecced to fit in your budget with an optical drive, SSD, and wifi card.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel - Pentium G4560 3.5GHz Dual-Core Processor (£63.47 @ BT Shop)
Motherboard: MSI - H110M PRO-VD PLUS Micro ATX LGA1151 Motherboard (£42.30 @ Novatech)
Memory: Crucial - 8GB (1 x 8GB) DDR4-2400 Memory (£55.41 @ Ebuyer)
Storage: ADATA - Ultimate SU800 128GB 2.5" Solid State Drive (£49.45 @ Amazon UK)
Case: Zalman - T2 Plus MicroATX Mini Tower Case (£23.99 @ Overclockers.co.uk)
Power Supply: Corsair - CXM 450W 80+ Bronze Certified Semi-Modular ATX Power Supply (£49.89 @ Ebuyer)
Optical Drive: Lite-On - iHAS124-14 DVD/CD Writer (£12.54 @ Ebuyer)
Wireless Network Adapter: TP-Link - TL-WN881ND PCI-Express x1 802.11b/g/n Wi-Fi Adapter (£11.99 @ Amazon UK)
Total: £309.04
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-24 20:43 BST+0100

Definitely cutting a few corners, but that's sort of a given with such a small budget. You would need to verify with the retailer that the H110 mobo comes with a BIOS new enough to support Kaby Lake.