The 1660 will perform better in most situations while using a lot less power.
Look up some reviews between the two, given the price difference is fairly small. You should be able to find a 580 or 590 for around the £150-170 range though, and as I mentioned earlier a RX580 will perform close to the 590 once you start playing with OC'ing. I'm assuming you plan on playing at 1080p, as the cards you're looking at are best suited to that resolution. That being the case, the 6gb on the 1660 will be fine.
My basket at Overclockers UK:
Total: £504.97 (includes shipping: £0.00)
An example of what I mean by pricing^
I'd take any of those over a 1660 at those prices, but when the difference is £15 if you "must" buy from that specific shop the 1660 is probably the better option for you.
I'd actually recommend contacting the store and asking them if they'd be willing to price match, tell them that if they're not that you'll be buying the GPU from elsewhere. No harm in trying your luck, they might be happy to do so given the money you're spending.
Edit: Added a Gigabyte 580 to the list, worth noting that while it's more expensive it has UK based customer support and an extra year manufacturer warranty.
Edit2/PS: A single fan 1660 is fine, they don't generate a lot of heat compared to the 580/590 cards.