I think schools will not open on the 1st June for a few reasons.
A couple of evenings ago Hancock said to a reporters question that schools when they go back would go back as they normally do.
People suggesting a phased return are not thinking how unlikely it is to help on a regular basis.
If classes are reduced and one year groups go in separate days then families with more than one child will still need to be looked after at home. Some families may have children at different schools which do things differently so not providing a regular schedule for return to full time work.
Also wrap around care will be unlikely to be on if Schools go back on a rota basis as it would not be feasible for cost and also not enough staff to safely social distance kids in wrap around care.
Hours of school 8:45 - 3:15 is this going to help with parents who have to go to work?
So if like Mr Hancock is correct and schools are to resume normal operations I feel it will most likely be September as this will give the government the time they are looking for to try and reduce the chance of a 2nd peak.