Overwatch favors nV cards, though we have yet to see if that's true with the RX480 or not. Anecdotally, we've seen some streams of people playing with RX480 review samples and it looks buttery smooth at 1440p (and one guy even claimed 4K...), so it would *seem* that 1080p high refresh should be doable.
Considering a GTX 980 gets close to 140 fps average @ 1080p Ultra and a RX480 is expected to be around that performance level, maybe higher once OC'd, it would seem like the best option for your desired quality/performance levels, provided the peformance pans out as expected.
Custom cards are expected mid-July so we'll have to wait on those to see what typical overclocks will run on a RX480.
Battlefield 1 is too much of an unknown quantity at this point, though traditionally the Battlefield games have run quite well on AMD cards.