Ok, it's a bit over budget, but anything under that would be wasting your money.
PCPartPicker part list:
https://uk.pcpartpicker.com/list/TwnQtJ
Price breakdown by merchant:
https://uk.pcpartpicker.com/list/TwnQtJ/by_merchant/
CPU: AMD - Ryzen 3 1200 3.1GHz Quad-Core Processor (£86.99 @ Aria PC)
Motherboard: Asus - PRIME A320M-K Micro ATX AM4 Motherboard (£53.86 @ Amazon UK)
Memory: Kingston - FURY 8GB (1 x 8GB) DDR4-2133 Memory (£76.97 @ Amazon UK)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£38.34 @ Aria PC)
Video Card: EVGA - GeForce GTX 1050 2GB SC GAMING Video Card (£101.94 @ Aria PC)
Case: Zalman - Z11 NEO ATX Mid Tower Case (£64.43 @ More Computers)
Power Supply: Corsair - CXM (2015) 450W 80+ Bronze Certified Semi-Modular ATX Power Supply (£48.53 @ Box Limited)
Optical Drive: LG - GH24NSD1 DVD/CD Writer (£12.99 @ Amazon UK)
Operating System: Microsoft - Windows 10 Home OEM 64-bit (£82.74 @ Aria PC)
Wireless Network Adapter: TP-Link - Archer T1U USB 2.0 802.11a/n/ac Wi-Fi Adapter (£14.00 @ Amazon UK)
Monitor: AOC - G2260VWQ6 21.5" 1920x1080 75Hz Monitor (£97.97 @ Aria PC)
Total: £678.76
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-11-11 20:57 GMT+0000
By the way, you could consider using the unactivated Windows 10? You just get a silly watermark and the lack of customization options. That would add $90 to your build, maybe to use in more RAM or an SSD.