If your contract says one month (which most school support staff contracts will say), then it is one month, regardless of when that month falls.
A couple of years ago, I was working in a school support role - I was looking for a non-school role, and was offered my current job on 6th August that year. I emailed my notice to the HT that same day (and copied my line manager into it) - HT responded the same day with a very sweet, yet professional email, confirming that she had received my notice.
I had to work until 6th September (as per my contract - one month).
With return data after the summer holidays, it worked out that I had to return to the school for 3 days in September, which I gladly did (it was essentially just a detailed handover with my line manager, and I produced documentation in that time for my replacement).
I've spent many years in schools, and support staff hand in notice during the holidays all the time - some return for a few days (like I did), some don't come back at all (if they handed in their notice at the end of July/before 1st August). I have also known HTs and other senior staff, to go into schools over the holidays, to conduct interviews if needed.
Good luck with whatever you decide to do OP!