My wife works at a Eton group London public boys school and their selection and interview at 11+ for boys from Primary are 'different' than those from a selective Prep. There are certain expectations from selective preps that wouldn't be there from primary state schools. The idea is to try and give each applicant (private or state) an equal assessment.
At the university I teach, there is a huge in-balance of private school kids, when you consider they make up 7% of kids at school nationally, The Sutton Trust says pupils from eight schools filled 1,310 Oxbridge places over three years, compared with 1,220 from 2,900 other schools. EIGHT schools- guess which ones? Well most are private and they include the schools that dominate the chat of tiger parents on mumsnet- St Pauls girls, St Pauls boys, Eton, Westminster , Kings College etc.
So I wouldn't worry about 'bridging a gap' so to speak, any selective 11+ school is very familiar with the differences, they will be looking for potential in your DC.
I would familiarise myself with the entry route for your hopeful school now, speak to them, explain your situation, don't be afraid to engage with the school. Check out bursaries and work out how are you going to afford fees and extras , the fees quoted aren't the only thing you'll be spending sums of money on-clothes, trips, music lessons, yoga lessons!
Some parents invest in tutors for exam entry, my advice would be supplement areas with past papers yourself, especially coming from the State sector the school will view your DC outside the hot house conduit of selective preps.
I would encourage after school activities- especially mastering a musical instrument, which encourages practice and commitment, and culturally enriching your child with museums and art galleries, concerts and encourage an interest in current affairs. Debate with your child and books and learning should be everywhere in your home. Its hard to instill these things if you don't set an example yourself.
Do these things and your child will be at no real disadvantage, especially as he/she will be assessed as a state school applicant.