STR:
Actual result
The user remains at the bottom of the page, unable to see the success message that confirms his account update
Expected result:
The user is redirected at the top of the page after he clicks on Update my Profile (this is how it functioned on the legacy pages)
Notes:

I believe the legacy frontend reloads the page when updating the information, hence the "move to top". I guess we could use the line below in the UserProfileEdit component to scroll to top:
ReactDOM.findDOMNode(this).scrollIntoView();
Do we want to do that?
I would prefer we just implement #4998 instead.
@bobsilverberg #4998 won't cover the scenario in which the user receives an error after submitting a profile update:

I would like to work on this issue
I noticed the presence of https://github.com/taion/react-router-scroll in our dependencies, maybe we can use it.
@tsl143 this issue is available if you wish.
Verified fixed on AMO dev with FF60, Win10x64 and Android 7.0
I've also logged #5250 which looks like a regression

Most helpful comment
I would prefer we just implement #4998 instead.