I am in the same boat, relocating and MK is one of the areas I am considering. I called school admissions yesterday and there are very few schools with Year 1 places, I was looking at Shenley Brook and I was told that nearest school with a place was 3 miles away and it is not that great. The waiting list for the local schools aleady have 15 kids within catchment areas so do not hold out much hope of getting into one soon.
Apparently the have not built enough schools to cope withthe demands of th new housing stock so houses are beautiful but scholling a problem.
The School admissions were very helpful
Other year groups may be different so you may be lucky.
I am considering private or going a bit futher up to Northampton. Hate the thought of putting her into private education at such an early age as it is important for her to have the vast social interaction at this stage.
Totally sympathise with your situation, relocating is stressful enough and I did not expect schooling to be such an issue.
Please let me know how you get on and good luck with the move.