For the Y2 child, i don't think it will be a big issue for them or for the school. Unless the school is an Infant school or a First school, the Y2 SATs results are not their 'main assessment / accountability' point, and in fact progress is measured from the Y2 SATs, so sightly deflated Y2 measures are no disaster for an all-through primary. Also the whole process is MUCH less formal so not being prepared will be less of an issue for your child.
The Y6 child is more complex. Are they going back into a school they have already been at? Will most of the others be going on to the same secondary? How big is the school? (the reason size will matter is that in a 1 form entry, each child is 3.3% of their results. In a 3 form entry much closer to 1% so less significant in terms of league tables etc). Y6 SATs - especially the Maths and SPaG - are tests it is useful to prepare specifically for, andf they are high stakes for primary schools.
If I were you, and the Y6 child wants to go back to school to meet up with established friends, I would start back at school on Monday 20th May. SATs will be over, but there will be a couple of months of term left - plenty of time to re-establish friendships and make new ones. However, there won't be much academic work going on in many schools: lots of 1 off events, residentials, mock trials, debating, school plays etc etc etc, so if you feel he is 'behind', you may prefer to ciontinue home ed until the end of the year.
The only thing you would need to talk to the school about is writing assessment. Usually, this is based on teacher assessment of writing over the course of the whole year, but the deadline fort this assessment is later than the papers IYSWIM. I don't know whether a child entering the school on 20th May would need to have a writing assessment result.
Does the school have space for both children on the required dates??