You don't.

Your wife doesn't by virtue of her US residency which she should carry the requisite evidence of. This assumes she remains airside. Else she will need a visa. I have heard that this is not strictly enforced if you have a flight departing same day but from a different airport, eg arrive LHR, depart LGW. But this is annecdote.

I suggest you print out a copy of the rules from www.ukvisas.gov.uk just in case the check-in staff aren't familiar with the rules.