Hi there, please don't panic! Your little baby is only 7 months! When I first read the title of your post I thought maybe you had a child turning five in the next few months or something!
Just wanted to agree with localbatterseagirl. Don't panic. My child is May born and we started to look round schools that summer, a few more in the autumn ahead of doing our application through the local authority in the January, then she got her place in April and started in September. It was all fine. That was similar to the vast majority of our friends, but even a few who have gone for fee-paying schools didn't start really looking at schools till their kids were 2/3.
Obviously it depends on what you want and I'm sure some of the independent schools do have very long waiting lists (although I'm also sure lots of parents apply to a lot of schools too, which puts the waiting lists in context); but I agree with another poster that they are businesses and perhaps it's in their interest to make you feel it's urgent even when your child is teeny tiny. I'm just saying please don't worry too much!
Best wishes to you and your little one!