Welcome to the 90s.
Here's what I'd do to save this batting lineup. It needs strengthening as a matter of urgency. All this talk of DBD, Livingstone is all well and good, but I think an experienced player at the international level would be a better option. Hales' test record isn't that far from Moeen's, and I'm pretty sure he'd be doing a better job than Cook at the moment, even if he's a muppet.
Hameed (Cook's replacement, time for him to return)
Stoneman (He'll come good eventually)
Cook (try him out lower down, otherwise time for retirement)
Root (No questions)
Roy (He's shown he has what it takes against the best ODI and T20 sides, and he's a far stronger option than Vince/Ballance etc. He's experienced at the international level now too- he won't be as green as the youngsters. Less time to adapt).
Bairstow (No questions)
Stokes (No questions)
Woakes (No questions)- Given the current circumstances is virtually worth it as a batsman
Broad/Wood/Plunkett- Needs to be a variety bowler
Leach
Anderson