You'll find a lot of city workers are ex army etc. This will be a help to you as the banking and finance sector appreciate the skills you would have learned in the army. Regimented, team player, working under pressure etc. As for doing engineering, maths etc. if you put your head to it and put in the work you will get there - it's 99% putting in the time (I know from first hand exp). I'd recommend starting with a GCSE in maths before an access course. Even in business it will pay dividends and what is an extra year if you are working towards something you want. Note: It's easy to mess a degree/access course up from poor time management /procrastination alone regardless of the subject. I've seen many people mess up this way that you would consider highly intelligent & those that didn't have it naturally and put in the work thrive - you probably don't need me telling you this but in my opinion that's the secret to doing well regardless of intellect (within reason obviously).
Re University - it depends on where you are based but any red brick type will do the job. If you are in London then Birkbeck has a great reputation / access courses and is geared towards older students with evening study but watch out as it's like the SAS of study, I'd say on par with LSE - you need to be very self disciplined and able to put in the work yourself to do well there otherwise you won't last at it but in terms of teaching a lot of the professors are old Oxbridge, LSE types that do world class research etc and its top quality in that regard plus its known in the city. If you are up north then Durham has access courses etc, Durham tend to be very good in their access courses and there is lots of help for those that struggle. Being 32-35 is not going to make any difference in terms of getting a job. Your motivation/ personality at interview/ degree result / interests matter more. It's a meritocratic type area to work in (on average) they will appreciate that you have made so much effort to go back and study later on (which is always harder).
In terms of the apprenticeship scheme with Barclays I made a slight mistake in that it's more of a internship scheme with the major banks but can be done without a degree (check out Barclays AFTER program). When it comes to learning about the industry there is no substitute for putting in the research, start reading articles books from reputable sources is the only way and avoid courses and websites that claim to teach you trading in a week/month/year etc. If it's too good to believe then run a mile - there are so many scams out there that your better of becoming well read around the subject before you go any further. For becoming more clued in on the industry in general read Michael Lewis books or the Market wizards ones and reach out to people working in the industry / your field of interest through Linked in etc...
Some stuff for you to check out online-
High frequency trading, algorithmic trading, quantitative trading.
Top down analysis, technical analysis, fundamental analysis.
D. E Shaw, Renaissance Technologies, Sorus fund management, Tiger management corp
Then if you find long term investing interesting check out -
Warren Buffett - Berkshire Hathaway (Best businessman in the world bar Jeff Bazos)
Benjamin Graham - The godfather of value investing (inspired most that invest / trade money)
Bill Gates - Not widely known that he invests money influenced by Buffet/Graham
John Maynard Keynes - Not widely known but he was a successful investor too
Carl Ichan - Another Investment tycoon graham follower
Seth Klarman, Dr Michael Burry, (Graham/Buffet followers)
Then you have different types of areas such special situation investing
Joel Greenblatt (Graham/Buffet disciple)
Fixed income/Bond trading and investing
Mike Vranos
Bill Gross
Debt Investing
Wilbur Ross
David Tepper (to an extent)
Macro Investing
Ray Dalio
George Sorus (Forex investor too)
Growth investing
Philip Fisher (Management focused)
Famous english investors.
Neil Woodford (Graham/Buffet disciple)
Anthony Bolton (Graham/Buffet disciple)
Crispin Odey (Graham/Buffet disciple)
Terry Smith (Graham/Buffet disciple)
Guy Spier (Graham/Buffet disciple)
This should be enough to keep you going for a few years lol. You'll notice that a lot of Graham and Buffet followers, this is because trading and investing is inherently very risky. The vast majority of investors and traders loose money and those that don't know or follow systems that have been proven to work over the very long term (From the 1900's). If you look at academic research then "Value Investing" which was invented by Benjamin Graham in response to the 1930s crash is a as good as proven system during booms and crashes. In the US. Graham went on to teach Warren Buffett who is now the second richest man in the world and many others at the time who are not as well known.
As far as human / computer trading goes there are not as many solid examples of systems that have worked over the very long term. Statistical trading done by the likes of D. E Shaw and Renaissance Technologies are the exception to the rule but even the founders mention Buffet and Graham a lot which is suspect. Interestingly Jeff Bazos worked at D. E Shaw before starting Amazon, even he mentions Graham and Buffet a lot. You might be seeing a pattern emerging here.
The reason I am telling you all this is that when I started I made a lot of mistakes, was interested in trading and technical analysis first but then moved swiftly along to Value investing when I got more informed. I lost a lot of money trying to trade in the early days (banks and large trading institutions have a huge informational advantage that the individual can not replicate) Computers are now so advanced that high frequency trading is done by fractions of seconds and the field is slightly corrupt - Search for "front running".
Once I moved to value investing after reading practically everything I could on the subject my results changed. The only downside to this is that you've got to do some really hard work. You've got to motivate yourself to learn about accounting, finance, industry specific trends, cyclic businesses, management ability etc etc. You practically have to become a hermit librarian style geek but the result is that you will make money and lots of it over a long period time. If you can show passion and interest you'll have no problem getting a job for a boutique type investment house when you qualify and investment jobs take from all disciplines whereas trading hires more mathematical types as i mentioned in my previous post.
Regardless of all I mentioned - when you get on the job you have to do a lot of your own research / analysis. Traders are practically left to their own devices from day one. Even Goldman Sacs have a so called training week/month in the US for new recruits - they do nothing more than show you how to operate the trading desk and you are left to your own devices afterwards. I am of the opinion that its bad to loose your own money but worse to loose others. The stock market is a big casino for most participants and to survive you need a solid system / plan.