I agree with everyone else, most schools, even those over subscribed at S1, will have spaces for catchment area pupils at S5. But you need to move up in May as S5 starts in early June and timetables are made up in April.
If I were you I’d rent up here first and move up as soon as she has sat her last GCSE.
State schools won’t accept an application until you have your new address.
Good properties in the catchment areas for good schools are selling within a few days ATM. Most will go to closing within 10 days or so and often stop viewings before that if they have enough notes of interest.
You can’t do this from Surrey.
Most estate agents up here won’t even let you view ( even if you could manage the logistics ) if you are still to sell in England.
Even if they did , the vendors solicitor will advise then against accepting your offer unless they want to wait 4-5 months and then have you pull out .
You’d usually be expected to complete here in about 6-8 weeks.
Oh and you must MUST use a solicitor based in Glasgow . Do not even think about using anyone in England , even if they are dual qualified . You need someone who knows the local market. Even more important as you don’t.
Good luck.