If SAG strikes, just about every scripted production worldwide will shut down, not just the ones WGA is able to picket. Shows that are in post-production will continue finishing, but at some point the pipeline of new films will run dry and even post will have to pause for a while. There are some that actually think that a SAG-AFTRA strike will actually speed things along, since the studios can't completely kill off production or they will suffer a shortage of content during future quarters.
There is also a thought that with July 1st brings the possibility of the start of force majeure, where studios can legally exit contracts for shows they no longer want to carry. Some people think that's what they have been waiting for and once the shows they wanted to get rid of are cancelled they'll come back to the negotiating table. But there are others who say that's just a rumor.