Will Mew leave when Celebi arrives?

Maybe

At GoFest, people were running both Special Research at once.

There was a Niantic support response that said the Mew questline will never disappear for anyone until they have completed it.

They've been wrong before, but I'd like to think they're right on that one.

I doubt it would go away. The asshurt from people stuck on either the Magikarp or Ditto quests would be crazy.

Unlikely that Mew quest leaves, they already demonstrated they have the code for doing both simultaneously at Go Fest. There's literally no reason for them to anger players who haven't completed the Mew quest yet.

No. Niantic already confirmed this some time ago.
These quests don't expire. You can complete both at the same time as it was experienced for some people at Go Fest.