I recently got a laptop with a Atom Z3735 processor and 2GB RAM (an Asus X205TA), and here's what I've found:
In Internet Explorer using Youtube's HTML5 player, I was able to play a 60fps 1080p video, and it certainly looks smooth as silk to my untrained eye. Unfortunately, I could not get Youtube's "Stats For Nerds" feature to actually display the FPS; it just displayed a dash. If someone knows how to get it to display FPS or dropped frames count in Internet explorer please let me know.
In Google Chrome, there were a couple of dropped frames, but not very many - I would estimate about 3-5 per second, for an overall framerate of just over 55fps - very close to the full 60.
Based on these results, I would fully expect the Intel Compute Stick to be able to do 60fps at 1080p. My general experience has been that Internet Explorer processes video more efficiently than Google Chrome does (simply based on the fact Internet Explorer could play videos that Chrome could not on older hardware), so I wouldn't be surprised if Internet Explorer reached the full 60fps. Also, playing Youtube through a web browser tends to be less efficient than playing through a dedicated Youtube app (which the Compute Stick would presumably be using), so if you take that into account I think reaching the full 60fps should be no problem for this chip.
Someone correct me if I'm wrong, but I don't think GPU is very relevant in this case because the current Atom line processors are System on a Chip (SoC), so the GPU is integrated right into the processor chip.
(And FWIW, I'm honestly quite surprised at how snappy this laptop is, considering the price and relatively low specs. The Baytrail Atoms have certainly come a long way since the Atom line was first introduced.)