How long will flights be shut down? I will be flying British airways from California to UK this July. I hope restrictions won't occur?

No one knows when travel restrictions and social distancing recommendations will be lifted.  They're all playing it by ear.

Even with a near total shutdown where I live I see about four commercial planes flying over each day. So some travel is still happening. I'd guess you'll be okay. You just may have to sign your life away to indemnify the airline in case you do get sick. I'm in California btw.

Flights are not shut down. Planes are still flying. The issue is whether the UK will let you in. You can try looking up what restrictions are in place for entering the UK right now. The US is not allowing travel from Europe, except for US citizens, so your citizenry matters, too. 

hopefully it will open up soon