We first encountered Moira MacTaggert in the 2006 X-Men: The Last Stand. The movie takes place sometime in the "near future" of the 2000's.
She appears to be a 30-something year old doctor on Muir Island. She is played by a then-36 year old Olivia Williams.
In the 2011 X-Men: First Class, we meet Moira MacTaggert again. Here, she is 30-something, but this time is an American CIA agent in 1962. She is played by a then-32 year old Rose Byrne.
The X-Men Movies Wikia claims that these two characters are the same person in-universe.
Sometime after the events of First Class, MacTaggert left the CIA and ultimately became a medical scientist and doctor. She establishes her home and workplace on Muir Island in northern Scotland, and eventually works alongside Charles Xavier again — this time in an educational capacity.
But they provide no citation or explanation how the character would span the 40+ year time gap without aging.
The Days of Future Past timeline re-write is not likely a good answer here, as we catch up with Moira in 1983 during the events of X-Men: Apocalypse, where she is still a CIA agent.