There are number of things that will affect your application.
The first is how big your chosen schools bursary pot is. What exactly does it say on their website? These are taken from 2 schools known to be generous with bursaries. "We are committed to broadening access to the School by offering eligible parents means-tested financial support in the form of bursaries for up to 100% of fees payable." "Each year there are funds available for free and subsidised places. The school benefits from the diversity which our pupils bring." In both cases generous bursaries are available unattached to any kind of scholarship but the reality is that this is rare. These are big names with lots of assets and correspondingly high fees. But most schools will attach any kind of bursary to some sort of scholarship usually academic.
So firstly you need to read the schools website and find this out next you need to have an open an honest conversation with the bursar (he's the man or women who controls the money in most school with a governing body the head does not do this) he knows how much is in the bank how much of that is available for bursaries and also assess the individuals financial circumstances. You need to work out how much you can realistically afford lets say its 40% of the fees you then need to speak to the bursar say looked at the prospectus/website love the school your think your DC will flourish etc etc at the school but you would need a bursary of 60% and do they or have they ever awarded bursaries of that size? Your not a asking for a definite yes or no just some idea if it is worth visiting it, bringing your DC, sitting the entrance exam, marvelling at their wonderful facilities oohing and arrhing over their fab exam results etc etc all a complete waste of their time, your time and unfair on your DC if their hopes are going to be raised then dashed if they never offer a bursary of more than 30%.
If you get the green light then visit the school preferably if possible meet the head emphasise what your DC will bring to the school; incredibly brainy, UK judo champion the youngest child to get a grade 8 on the oboe don't slag off your current state school you've got to be offering the school something they don't really care that your DC is one of 35 and the teacher cant add or even spell Wednesday properly!
If it still looks positive you will most likely be asked to complete a registration form and sometimes pay a registration fee and also at some stage a bursary application form.
IME those who are unsuccessful and have done what I've suggested and had a positive feed back from the bursar have money in some shape or form; 2nd houses, assets, shares, savings, trust funds, lots of equity in their current house, a Monet on the wall of the garage. Many on MN bandy around the figure £40 000 maximum earning we earn significantly more than that and still qualify for a bursary a significant bursary. They will expect both parents to be working where possible especially if all children are of senior school age although caring for elderly relatives (could become a problem for me) disabled children etc will usually be taken into consideration. Most will look on google earth nowadays to check your not secretly living in a massive pile, the bursar may even visit you at home to check the Monet isn't real and turning up in a brand new range rover is also unlikely to go down well! Many say the forms are intrusive but so they should be many are hoping for a a significant reduction and therefore they should expect their finances to be carefully scrutinised. We don't find them too difficult as we write can honestly write 0 for everything; saving shares trust fund (in my dreams) etc etc the hardest thing we found the first time round was locating our tenancy agreement to send them a copy! We are not asked about our fuel bill food bill or council tax I think in our schools case an amount is allocated for this but we are asked about how much our other DC costs in terms of education and many will go into every micro detail. Many sadly complete all these stages and are still not successful but as one bursar said to me once if you don't ask you wont get. Be honest with your DC right from the beginning say it might not come off then if it fails it wont be a complete disappointment.