Alas not many jobs listing in Edinburgh at the moment. NannyJob: Edinburgh
10 Gross an hour is probably about right given the location (the agency can advise on what sort of salaries jobs they have successfully filled have been), which for 45 hours a week would be £25,600 ish once you add on employers NI. Then you have the other additions to add, such as activity budget, a bit extra in food (nanny will typically eat lunch with your children), mileage (if nanny uses their own car to transport your children).
So if you were to say £30,000 it should be within that and may well be less.
Activity budget wise - track what you spend for a while. You need to get a feel for if £15 a week is sufficient or if it is nearer £30 a week. When children are very young then the cost of activities if often quite low, as they like toddler group, get cheap entry to swimming pool, free travel on busses/trains. Costs tend to go up as the children get older, especially once aged 5+ when they often will not qualify for free/reduced cost entry to places, plus places they will go change - some will be happy running around the woods, others won't like that. Look in your area for what you have for your children now, and give some thought to future (look at the prices on info leaflets and see what age it starts to cost a lot).
45 hours a week is quite short for a full-time nanny in my view. As a part-time nanny I do 40 hours, a full time nanny could be doing 50-60 hours. So your job is quite attractive as long as the person does not need to be earning what they would earn in a 50/55/60 hour a week position. I can't imagine that living in Edinburgh would be that cheap, so you need to take into account what sort of costs the person doing the job would be having themselves - what sort of commute distance is viable, especially in winter.
Given your timings, nursery I suspect would be a lower cost option. Not by much though so the convenience of having someone come to your home may well outweigh the cost difference.
If I were to move to Scotland, I would be looking at jobs via agencies, so I think you are best going that route. There are not many agencies (as agencies in Scotland require registration, unlike in England) and they will aim to match applicants to the job. However it does come at a cost.
You could try yourself by advertising where agencies advertise - such as www.NannyJob.co.uk, local newspaper, local families magazine and you could use listing sites like Gumtree.com (which has an Edinburgh section) and Childcare.co.uk but you will be doing the advert design, weeding out whom to interview, everything really - so you need to consider if you have the time to do that, how much your time is worth.