s#s/his/her#s/his/their# also works and avoids awkward escaping. The first symbol after s is used as the separator. Works in vim, at least.

In other words:

s%s/s\\/his\\/her/s\\/his\\/their/%s#s/his/her#s/his/their#%

Did you just assume my regex engine is pcre gendered???

Wow, I've never heard this joke before. Original and well-applied to the situation.

    awkward escaping

Or as I've seen it called, "leaning toothpick syndrome".