I think it's easy:
- Vision says in Civil War "In the 8 years since Mr. Stark announced himself as Iron Man, the number of known enhanced persons has grown exponentially," i.e. enhanced persons have existed before, but more are revealing themselves now.
- The MCU already has division between people who do and don't trust enhanced individuals in general, but lets say people are more afraid of Mutants because they're often younger and have more dangerous/unpredictable powers that they're not used to.
- Characters all throughout the series use 'enhanced' to refer to anyone with abilities, regardless of how they got them. I'd suggest that 'Mutant' is like an offensive slur people use to put down enhanced people, hence why we haven't heard them be referred to as such.
Put it all together, and: Mutants have always existed, but Xavier's School has done a good job keeping most of it quiet. However, the increase in public superhero activity has meant more young mutants not hiding their abilities, hence the seeming increase in powered people. The word 'mutant' itself is slowly becoming adopted by enhanced youths, in the same way other marginalised groups in the real world will co-opt once-offensive slurs as a badge of pride.
The only little detail about the X-Men that might need changing would be Magneto's origin, because at this point to keep his time in a Nazi concentration camp, he'd either have to have been an infant at the time, or he's nearing his 100th birthday (probably both). Ditto for Prof. X; for them to have been friends as young adults, they need to be around the same age... Unless you do something ballsy and have that be part of the plot of the movie, that they're both super old and looking for heirs to their causes...