We don't know for sure
The film rights for Marvel's assorted characters are a complicated situation, especially when it comes to non-title, franchise-spanning characters like Uatu. To my knowledge, detailed information about the ownership of characters in that category has never been made publicly available.
There's some (admittedly scant) evidence to suggest his rights are owned by Fox. In 2014, BleedingCool reported on restrictions placed on sketch card artists related to Fantastic Four characters; they include an image of a letter allegedly sent to an (unnamed) artist forbidding them from drawing, among other characters, the Watcher (click to embiggen):
Transcript: (emphasis theirs)
All Marvel characters related to Fantastic Four are now off limits and will be immediately rejected by Marvel.
This includes but is not limited to:
This came shortly before Marvel announced the cancellation of the Fantastic Four title series, which is said by anonymous inside sources to be related to conflict between Marvel and Fox over the 2015 Fantastic Four film
I want to emphasize that this has not been publicly confirmed, and may be related to any number of other factors. It is, however, notable.