Depends what power supply you have? My gtx 750 is great for 1080p gaming(have a no HD screen currently) and I paid less than £50 used, but then I have a 300w power supply, if I had a proper case with atx psu support I'd have my 520w back in and something like a 270x again or a 660ti, 680 if one can be found, 760, 770, 280x, 290x if your lucky, 1050(ti) from the used market that uses at least 1 6 pin and is better than the 750(ti) models or new I'd go for rx460.
All will play a variety of demanding games at 1080p on high. Might have been a one off, but on gum down my way I saw a gtx 970 mini for under £100 fully working, so as you can find the full size versions for as little as £120 you might get lucky with one of those.