I was the typical age when I started my A-Levels - though I initially chose Maths and Physics - ended up completed failing Maths, while just passing Physics.
In my second year, I chose to continue the Physics - but basically self-taught the core syllabus to myself via a few 'concise' books and ended up doing okay in the end. - I would say it is very possible to 'self-teach' if you wanted - but it is expensive.
I ended up repeating about 4 exams for the Physics to change my overall grade (I think it was £15 an exam)- while not having the additional course costs, or other fees to worry about.
From personal experience you are not 'too old' for A-Levels, at my college there were plenty of twenty-somethings doing A-Levels - and for the evening classes mainly adults (thirties on-wards) attended.
Regarding the plausibility of it all - I do not think it would be 'too much' if you were to do not much else. A part-time job would be manageable, though I must confess I mainly just studied each night and had no other commitments.
However it will be expensive to self-study - I would say in your situation - you should weigh up which would be more useful to you.
A-Levels may offer you more - but will probably cost more - a loan could cover this - but could you / would you be happy to pay it back?
If the worst case scenario presented itself: i.e. just having A-Levels (no cadet scheme) or just having an Access Course - which would you prefer?
I know how you feel about Time - I feel the same - I wish I could take back years and say right - I will begin flight training at 16, 18 etc. But - that is in the past; what is important - as you say - is to do something now, while you can.
If I remember correctly I believe there may be some form of financial aid / bursary available to older A-Level / College education applicants? At-least, it seemed so were I studied?