@carebea I would start by looking at what private schools are local to you. Have a look on their website, usually under the 'fees' section there will be information on applying for financial assistance. All schools are different, some offer assistance from age 5, others will not offer assistance until age 12/13.
Once you have a short list of schools phone up and arrange a visit, do not mention that you will be applying for a bursary. All schools have different entry requirements, the younger the child the easier it is to meet the requirements, so ideally get them in as young as possible.
For most schools the entry process will start in the September for entry the following year. More prestigious schools will begin even earlier. When I have went to visit schools I always research the school fully, dress in business dress and make sure I talk well, shake hands on meeting everyone etc.
Most schools will invite your child in for a trial day. This can be tricky as all my children came home from their trial desperate to go. Obviously I couldn't tell them they could go and I did feel bad, like I was selling them a dream which was unrealistic. I dealt with it by telling them that I wasn't sure the school was completely right for them because of A,B,C, but said we would work out what was best. Next you have to register your child for the school, this is were you are usually first asked about applying for a bursary. Registration usually costs anything between £50 to £250.
You will then be sent out an bursary application. They can be daunting to fill out, but the best thing to do is fill it out as soon as you get it. It will ask things about household income, expenditure, assets etc. With expenditure include absolutely everything that you send money on. Even clubs which maybe cost £5 a week.
Your child will probably have to take the entry test before you know if you will get a bursary. Again, this can be tough, but I told them I still wasn't sure on the school but sitting exams gives good practise for the future. With one of my DC school I got a letter saying they had passed all entry requirements, and had been offered a place but had to wait another 3 weeks to find out if they had the bursary. I just didn't tell them they had passed.
One school offered 80% bursary at first which I accepted. I knew it would be tough paying the 20% fees but I was desperate to get them in. After a few months I was really struggling so I emailed the bursar and explained, he increased the bursary to 100% just like that. I really didn't expect that.
Another poster above said about children interviewing well, I completely agree with this. All my children have excellent communication skills and this has always been commented on at interviews. Make sure they know to look adults in the eye, shake hands on arrival and ask questions. They will usually asked why they think the school is right for them. I usually give them a few reasons why before hand.
Private schools love children who are effective leaders and will usually have some form of leadership game as part of their entry process. Make sure that children know that being a leader is about putting your point across but also listening to others. If they do a sport/music/art make sure the school knows about this, if possible brush up on skills before entry assessments.
If you don't meet the head as part of your visit to the school, phone up and ask for an appointment with them. Two of my children attend a very prestigious school and parents do not generally get to have a personal meeting with him. After my visit I phoned the school and asked for a meeting with him. His receptionist tried to fob me off but and it took three weeks but I eventually got an appointment with him. I sat and had a coffee with him and told him I was applying for a bursary and why I thought my children could benefit from the education the school offers. I also explained what the child would bring to the school. I think he maybe thought I was a bit mad, but at the end of the meeting he told me that although he couldn't interfere with the bursary application or process he would try his uttermost to offer them a place. At the very worst a meeting with the head shows your commitment to the school, your children education and they will hopefully remember you.
In my mind schools get more applications for bursaries than they can pay out. You need to find a reason for them to pick your child, keeping in regular contact, making it clear how very much you want you child to go there etc all help.