So I've spent the lat 20 years in a successful career that no longer interests me at all. Though it sounds crazy, I have always been interested in software development. My question is, given that my degrees are not in math and science, let alone computer science, is there any possibility of becoming a programer or engineer without going for a BA in computer science at least?
Can I ask what your speciality is at the moment?
I don't think this is going to work out for you. It's not the fact that you don't have the education (formal qualifications keep HR people happy, but they're not where most programmers get their skills from).
The big problem is that you're coming from zero, and up to this point you seem to have had little interest in the subject. There's a lot of sense behind the theory that it takes 10000 hours to master a discipline, and many good coders have put those 10000 hours in during their teen years. You're not going to have the deep background exposure to different problems and methods that most decent graduate-age programmers have.
Secondly, are you sure you want to spend your days doing that sort of detailed work? I started off as a programmer, but over time moved into management... as coding became a bit same-y and it became more interesting to 'hack' people. If you've spent 20 years doing people oriented work, you may find it very difficult to sit in a corner and code all day.
Finally, I think you'll have problems getting hired. Junior programmer to senior programmer/team lead to management from 20-35 or so is a standard career path... except these days a lot of low grade programming is outsourced (to India and the like). As a newbie low grade programmer appearing on the scene at 40yrs old (I'm making assumptions based on your 20 year career) you're going to be hard to manage... and you probably won't get hired by a 35 year old team lead. Ageist? Probably. Realistic? Yes. In my experience most older coders are contract staff with deep skills (and as a contractor, they're not in the politics or progression game).
A prospective manager will see you as bringing too much experience to the table to be a placid code-head, sitting getting stuff done and meeting deadlines.
Feel free to convince to stop thinking about this. However, if there is some practical advice you might have about actually making this change, I would appreciate it.
If you're interested in coding, why aren't you?
Spend a hundred bucks, join the iOS developer program, learn Objective C (loads of good resources around... the O'Reilly 'Head First' books are great; there's a free Stanford university iOS course on iTunes U in video). In a year's time if you've produced a high quality app that's getting some sales on the app store - and you're driven to program more, then you might have the beginnings of something. If you can't concentrate on it, you can't learn on your own or you just plain hate it - then you're only down some time and a couple of hundred bucks.
(Don't get too hung up on languages at this point. If you learn to program using a language that's reminiscent of C, then your skills will be transferrable).
Good luck!