So many questions. If you're not sick, don't go on sick leave, to me that's fraud. If you need the time off to look after your children during the holidays you can legally use parental leave but there are a mix of other things you could use for this if you're not going away for that whole time.
If you need the time off to, I don't know, project manage a house renovation, you can take a sabbatical, which would be unpaid I guess but you'd have to factor that into the cost of the build.
If you need the time off to do another job, like say working at a festival or some short term freelance opportunity, again you can ask to take it as a sabbatical or a CPD opportunity or something.
If you are actually experiencing burnout or a MH crisis, I'd assume that you're not able to schedule that for the summer because that's not how our bodies and brains work.
If you're not actually sick or stressed in any way, please please please don't you dare lie and take it as sick leave. It spits in the face of anyone who genuinely needs the time off.
This is why it's so hard to give any decent advice without the context.