Having Apple Watch series 3 (or better yet, 4, when it is released) will force you to go to iOS 12. But that’s a different story.
Yeah, I totallly understand. My point is that (IMO) if he upgrades his wife's iPhone6 to iOS11 so that she can use the AW3, they'll be buying a new iPhone very soon. Which they would likely be doing anyway. However, that iPhone6 would now be stuck on iOS11 and will be dog slow for his mom to use.
My suggestion would be to not even pass down the AW3, at least until your wife gets a new iPhone. Then the 6 can be passed down to your mom while still on iOS 10.3.3
Only a few more weeks to wait for the new iPhones anyway. Might as well wait.
Also, perhaps iOS12 will bring back some of the life that iOS11 has sucked out of older iPhones, at which point it would be safe to update the 6. However, matrix has been around here long enough to understand that anecdotes on Macrumors regarding performance varies greatly from member to member. Heck, there's lots of folks that insist (yes, I'm biased) that iOS11 doesn't affect performance on iPhone6 devices.
Aside - I'm gonna do an experiment after I get my new iPhone X2. Gonna get a new OEM battery for my iPhone 6, then run some performance tests. Take some videos showing how sluggish the thing is. Then wipe/reset it and run the same tests and take the same videos. I truly wonder if a full wipe will make the 6 run faster.