No way, May is far, far too early. I no longer have small children (mine all grown up) but I work in a non-teaching capacity in a school. It's impossible as it is to stop small children passing on coughs and colds and the virus will still be around by May. We are 'behind' many European countries in our 'peak' timescale, and our lockdown hasn't been as severe as in some, and therefore I think we'll undo all the good so far if schools go back too early.
I also think that when we start to ease back into normality, it should be slowly. Secondary schools should go back first as it's easier to segregate older youngsters, teach about social distancing and maintain cleanliness rules.
Primary schools should be much later. It isn't the pupils themselves who are so much at risk but the fact they may be carrying in virus from outside the school and/or passing it on to adult workers, teachers, cleaners, admin and other staff who will then take the virus out into their own families and communities again. Or the other way around, of course.
I would also want - in an ideal world - proper PPE for all workers in the school and for proper testing to be available. I can't imagine how it would work otherwise. I myself have had an odd illness...high temperature, aches and pains, red eyes, but only a small amount of coughing. I had it, took to my bed it was so bad, recovered; the following week I had the same symptoms reoccurring.
No idea if it was the virus or something else. Would I test positive or negative? Am I still positive? Would you want me to be dealing with your child when I don't even know? The whole thing is ridiculous and dangerous in my opinion.....