Well, pretty much everything in edinburgh is near to public transport - it's got a great bus system. Are you looking to be close to the city centre or further out or in a suburb? I can suggest some areas and good apps to search within if you'd like. Once a child turns three, they are offered 15 free hours of childcare a week, tho your local nursery may have a waiting list of those who've had their kids' names down for a while. But, in general, 3 and 4 year olds get 15 free hours of childcare a week, often at a nursery attached to the local primary school, often 5 mornings or afternoons a week.
As to part-time employment, be warned that childcare is quite expensive, between £38 - £45 per day. You may be able to use your 15 free hours in a private nursery, but I'm still learning how all of it works. You'd have to ask your eventual childcare provider.
The age cut-off for each school year is the end of February. Check out the edinburgh council website for info about when your 3 year old can start based on their birthday. Nursery=preschool.
Let me know if you have any more questions, happy to help.
