If you haven't already done so, it may be worth doing a search as I know there are lots of posts about Bromley and Bexley schools already.
If you are just talking about state (as opposed to private fee paying schools), the catchments for the good Bromley schools are usually very small and the house prices are higher because of that.
There are 2 super-selective state schools in Bromley, St Olaves for boys and newstead wood for girls. You're probably best to look at their websites with regards to in year admissions.
Bexley has a number of Grammar schools and again, its probably best to look at the Bexley Borough site re in year admissions.
The problem with all the good schools is that they are oversubscribed so I assume you would just have to wait until people leave and places become available.
Sorry, cant help re private schools, hopefully someone will be alone soon who can.
If you have any queries about specific schools, please feel free to ask, although as I said there are loads of past posts.
Good Luck