For some reason, I can't reply to the DM you sent me:
Here it is instead
I lose track of the years, so I've gone back and calculated (with reference to P6s I know): current P6s are the same age as your dd (I know some with birthdays in August, so only a few days difference in age to your dd). So she would definitely be going into P7 next year.
She wouldn't even have had the option of deferring if she's started school here as she would have been 5 when school started (mid/late August) so was legally obliged to start school - and anyway, that would've been highly unusual (in practice it is just the January and February birthdays - in your dd's example, it would be Jan/Feb 2010 birthdays - with just a few November and December 2009 birthdays and even rarer usually from abroad October.)
Don't call them public schools - that confuses people here (it used to confuse even me and I was brought up in Scotland
) - as "public school" = private school
- also called independent schools. You mean state school. 
In terms of showing whether or not she is a British national, no you don't need to. Any child resident in Scotland (and England for the moment for that matter) has a right to a state education. There might be an issue with children sent on their own from overseas purely for the purpose of education - but I think that that is more to do with further education fees.
In terms of providing evidence of living in catchment, you may indeed need to wait until June. Although I am vaguely aware that providing evidence that you have bought a house (the concluded missives) is acceptable (at least in some areas), so if you can provide evidence that you own the house - and perhaps the notice of termination to your tenants - that might help. By the same token, there is no minimum residence period - just that you are living there when you make the application.
I'm confused by your sentence l currently stay with relatives who pay council tax under their name, l just contribute to all bills. - are you in scotland now or are you still in HK? Not everyone pays council tax or will have it in their name, so there must be other ways of providing evidence.
If you move into catchment after applications have to be in for P1 entry, then the council is obliged to put you into the catchment school if there is space or a place in the nearest school at which there is space (and provide transport if it is beyond a certain distance). It might not be a school you like though. If that is the case and then a place later becomes available at the catchment school, then I think you have a right to get it (unless there is someone with a higher priority - eg disabled or "cared for").
If you are happy with the secondary school that your house is in the catchment for (or rather the catchment primary feeds), then whether or not your dd gets into that primary, she will be ok for the secondary school.
Hope that helps.