(I helped a lot when my DDs were in primary school.)
Stop worrying about counting and number recognition.
Just try to keep her happy over the next 6 months, and work on 'self care' which is more important than academics at this age.
It's going to be hard to practice playing with other children, but everyone will be in the same boat so all children will start 'behind' compared with usual reception years. Make sure you try to do some 'turn taking' games, and work on dressing, doing up coats, that kind of thing.
As an Oct birthday you actually have a great advantage, as your DD can be 10 months 'behind' and still be within class norms, just with the summer borns.
However, I'm not convinced the 'you can do this you are clever' is the best approach. I think many children don't like being 'tested' like this, and in general it is better to praise effort rather than 'cleverness'. I think you might have more benefit from finding a more 'playful' way somehow for this kind of learning.
You could always defer until January if you think she would benefit from some more normal time with you before school if the current situation continues well into the summer.
Finally, hopefully you have been able to pick a school with some understanding of issues that might come with being adopted. Prior to starting you need a discussion with the reception teacher and the senco on how how she emotionally reacts to being told off etc.
5 months is still a long time, she might mature a lot in that time.