[clangd] Fix race in FileIndex that sometimes temporarily lost updates.

Summary:
FileIndex was built out of threadsafe components, so update() didn't have data
races, but wasn't actually correct.

Reviewers: kadircet

Subscribers: ilya-biryukov, MaskRay, jkorous, arphaman, usaxena95, cfe-commits

Tags: #clang

Differential Revision: https://reviews.llvm.org/D82891