I would imagine they'd stick with Spotlight.
Sherlock of course.
Well I can't say because I can't say. But I won't use an Apple SE just because of brand loyalty.
Spotlight to find things on your Mac would stay.
For the internet, SearchLight perhaps?
I would use it if it was good, not because it be Apple's search engine.
Because "Results that are only what WE think you should be looking for" and "Anal retentive babysitter" has that oh so naughty "a" word in it, maybe "iDictator" would be suitable for Apple's regime, if they were going for the truth in advertising concept.
I'd call it waloshin. Then I'd design a cute little mascot and call him Wally.