Welcome,
Although you say that you do NOT want to join the RAF, I would suggest that you do look at the sixth form scholarship that they have on offer at the moment and see what it gives you, and in addition to this have a look at the Air League and GAPAN schemes to award flying scholarships up to PPL (though I believe it is Air League only that do this).
At the moment you are unfortunately in the same boat as many on this forum in that BA seem as though they may not be recruiting for about three years or so via the Trainee Entry Pilot (sponsorship) scheme, the age limit for which was 27 I believe until it was halted recently, not 21 as you heard.
1) What if you cannot get in at 18?
If they are not yet recruiting or you fail to get in at this age, you would obviously be silly not to look at university. Indeed many of those selected to train by the TEP scheme are graduates themselves so you at 18 will face competition from those possibly more life experienced and mature than yourself, consequently you should not feel bad about not getting in at 18, nor will it be hald against you.
2) University degree
You do not have to do an engineering or science based degree contrary to some myth, and my advice is to do what you are good at, be it economics, history, psychology etc, as all experiences have a place in the company or indeed any company: it all adds to the variety of people that you work with. Neither will advantage or disadvantage you in the selection, and even though for the licence exams a greater awareness of technical areas may help, the mind of any graduate should cope with the exams.
3) Join a UAS
You say you don't want to join the RAF, but give it a go and show an interest: it will give you opportunities you never would have otherwise experienced (if your eyes are perfect for pilot) and you will meet a multitude of different people, and it is far more than looking cool in flying suit and helmet. Indeed you may find that buzzing about at 250 feet in a fast jet is just tto addictive to refuse and end up there for 16 years, but I guarantee you will kick yourself if you don't try. And BA seems to love ex-UAS guys, due to the quality of the training received, and the rounded personalities. It is a commitment, and maybe too much if you are doing engineering or physics based degrees (though possible), so get in it in the first year and find out if you like it. Do the syllabus in 2 years and get streamed Fast Jet, then pressure is of for last year (or two).
4) How to best prepare yourself and improve your chances (and influence people: sounds like a Terrorvision album title!)
Well: get involved in things at 6th form and uni if you go that are based about teamwork, leadership and perseverance. Indeed they are many of the qualities that are looked for by top graduate employers so there again university will do you good. I was vice captain of xx team and I organised xx for the students' union are all plus points in the box. Be prepared for questions on 'why flying', and 'why BA in particular' along with keeping an eye on Flight International in WHSmith and perhaps Pilot, but more keep tabs on big aviation news: remeber that they look for 'manager' types in the selection who have potential and are aware.
With regards to flying, I would tend to not follow the advice of some above, and buy one lesson as a trial (Xmas pressie idea!) maximun, then save (or get somebody else eg RAF/Air League/Parents) to pay for the PPL in one go in a vacation. Why? Continuity matters and so do your savings if you go to university. More flying does not matter for BA, though a PPL is nice for general awareness of why you want to do flying etc. You will spend more and take longer if you fly just twice a month as opposed to the lot in 3-4 weeks, which will give you more confidence and help you remember those skill learnt. Certainly you should NOT try to build hours for the BA TEP selection, just do what it takes to keep the licence valid, and whatever UAS gives you if you get in. HOURS DO NOT HELP (unless they are all military training possibly).
With regards to the Direct Entry Programme (DEP): you cannot really build 2500 hours, as you are required to have flown a 25 tonne or greater transport aircraft or a military fast jet, so I am afraid you need to get a job in another company first! The age limit for this is in the region of mid-forties, rather than 28.
How to find money if you do build hours to get in a low-cost/regional?...Graduate from a good university and work in the city. There are jobs with starting salaries of £45000 not including Christmas and Summer bonuses, so working mad hours for 3 years would give you enough to pay for a full licence in theory. Otherwise a more moderate salary at a city institution or IT consultancy and wait longer or fly at weekends, or even a normal job and fly every other weekend.
Get good A-Levels/degree and you will be made, eventually getting into flying if you persevere.
Good Luck,
Lucifer