Your daughter still can go to P1 this year at least for state school. For private school the admission process is completed now and you normally need to have an assessment to get a place during Jan for the semester starts from Aug. Or you can put her on the waiting lists. If you plan to stay in the Scotland it’s ok to choose a good public school but if you are an expatriate it’s better to choose private school as they run both Scottish and international curriculum.
Yes, you need reg for your little one for the nursery, again good nursery has a very long waiting lists. the cost is about 1K per month.
BTW , if you both working full time, it maybe better off to hire a nanny rather than go to the nursery as the school normally starts around 9 and finishes at 1515, you would need before/after school club arrangement.