I’m sure he will be fine, but I wouldn’t underestimate either the difficulty or length of those transition ‘months’. I’d say don’t be put off, because kids are generally more adaptable and robust than we think - but equally be prepared for some bumps along the way. Our kids were much younger - 8 to 12 - and the younger two found it much easier. 10 years later they’re still more French than the one that was 12 when he arrived (he still prefers France though).
If the 16-year-old is going into school be aware that this is a tricky age. Many French schools are pretty rigourous - not to say rigid - by this age, especially if he is academic and wants to do a bac général. Language support varies a lot. Our regular expert on French schooling is @vero , who is French and teaches at this level.