If you are considering changing to Catholic, our local catholic school gets kids bussed in from miles away and you need to provide documentation for baptism, 1st communion and confirmation and various other bits from the local priest confirming regular attendance and such like, so it's not an easy option unless you have some Catholic roots in the family to draw on. Might be easier to move house!
Private is OK if you can afford it, however, if circumstances change and you have to pull the kids back to local schools, you'll be at the back of the queue and only get the left over spaces after all the prime spaces have been filled.
We were fortunate when we applied for our eldest as the closest school which we didn't want the kids to go to was closed meaning the school we'd have chosen actually became our closest and now we get the other 2 on the sibling rule anyway. We were also fortunate in the fact that the school is one of 2 that managed to retain their building projects, so we're part of the way into the rebuild which will be completed for all of ours to do their CGSE's in brand new classrooms.