Has Hollywood started filming TV shows and movies again?

Film production will resume in the UK, under restrictions. 

The films include:
*Mission Impossible 7 
*The Batman 
*Jurassic World: Dominion 
*Fantastic Beasts 3 
*Sony’s Cinderella musical
*Disney’s The Little Mermaid remake. 

https://www.thewrap.com/mission-impossible-7-the-batman-production-resume-uk/

I do not think so yet.

No. There are a few TV shows that are doing at-home or zoom shows, like SNL, The Tonight Show, and Good Morning America. Nobody's back on set, though, neither for TV nor movies.