The defining thing about the independent sector is, well, that the schools are independent and can therefore, subject to a few constraints, decide for themselves what sort of scholarships to offer, if any. The schools themselves vary so much it's impossible to generalise, but I think very few prep or junior schools would offer any sort of scholarship provision to the youngest children.
For example, Eton, as well as being a school - duh - also happens to be one of the UK's biggest landowners, with a rental income to match. It offers academic scholarships, which could if necessary, I imagine, be topped up with a bursary (ie. a means tested sum of money unconnected to the scholarship, which would be awarded for academic promise).
Schools obviously vary in how academic/musical/sporty they are; their scholarships are awarded accordingly. So, a few years down the line your little Freddie is a real bright spark but can't play football to save his life. Would prefer to pore over his Latin textbooks, or do a spot of extra geometry; the academic creme de la creme for him would be a scholarship at Winchester, St. Paul's, Westminster etc. A scholarship from a school considerably further down the league tables (hate that term) would, in the academic hierarchy, be "easier" to get, but could be just right for your family. And of course there are awards for art, music, sport, good all rounders, etc.