Hi OP, Wirral mum here, both Wirral and Chester have nice and not so nice areas. As people have previously stated Wirral is a grammar school area.
The go to Catholic school is St John Plessington (rated outstanding) located in Bebington, this school is oversubscribed so it is likely that if you wanted an in year transfer you will probably have to appeal to get a place as they are currently over their published admission number for the new year 7 that will start in September. The executive headteacher is also mentoring in another catholic school on the Wirral called ST Mary's Catholic College (reqires improvement) located in Wallasey but it will a few years until that school is at the current standard of SJP.
Wirral Grammar School for Boys (rated outstanding) located in Bebington, is currently under subscribed for the new year 7's and the school has put out a message asking prospective parents who's sons have not taken the 11+ to get in touch with the school to see if they can arrange them to sit the test.
Calday Grange Grammar (rated good) located in West Kirby, is in a beautiful and affluent area of the Wirral and has a huge "catchment" and they currently provide school transport for boys for North Wales and Chester. I believe they also have available places but you would have to phone the school directly to confirmand. The school has a friendly feel and after an after a few issues 5 years ago the school seems to be going from strength to strength.
St Anselms (rated requires improvement 2016)located in Oxton, is the only Catholic grammar school for boys on the Wirral. Despite the Ofsted this still appears to be a popular school and still has people appealing to get in each year.
Birkenhead Independant School - Is a popular private school located in Oxton. It is co educational and has a good local reputation.