I would say AMD would be a better option for you to game and stream at same time.
PCPartPicker part list:
https://uk.pcpartpicker.com/list/Q6wTcc
Price breakdown by merchant: https://uk.pcpartpicker.com/list/Q6wTcc/by_merchant/
PCPartPicker part list:
https://uk.pcpartpicker.com/list/wjzzTH
Price breakdown by merchant: https://uk.pcpartpicker.com/list/wjzzTH/by_merchant/
Also the 580, like the Red Devil Golden Sample is faster than 1060.
CPU: AMD - Ryzen 5 1600X 3.6GHz 6-Core Processor (£214.99 @ Amazon UK)
CPU Cooler: Cooler Master - Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler (£26.48 @ Novatech)
Motherboard: ASRock - Fatal1ty AB350 Gaming K4 ATX AM4 Motherboard (£97.98 @ CCL Computers)
Memory: Kingston - FURY 8GB (2 x 4GB) DDR4-2400 Memory (£64.99 @ Amazon UK)
Storage: Samsung - 850 EVO-Series 250GB 2.5" Solid State Drive (£85.96 @ BT Shop)
Storage: Western Digital - Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive (£39.95 @ Aria PC)
Video Card: PowerColor - Radeon RX 580 8GB Red Devil Golden Sample Video Card (£318.87 @ Amazon UK)
Case: Corsair - 200R ATX Mid Tower Case (£56.56 @ Ebuyer)
Power Supply: EVGA - SuperNOVA G2 650W 80+ Gold Certified Fully-Modular ATX Power Supply (£84.98 @ Amazon UK)
Optical Drive: LG - GH24NSD1 DVD/CD Writer (£13.84 @ Amazon UK)
Wireless Network Adapter: Asus - PCE-AC56 PCI-Express x1 802.11a/b/g/n/ac Wi-Fi Adapter (£47.98 @ Ebuyer)
Total: £1052.58
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-07-29 22:29 BST+0100