Tim,
I recall those days in my youth. The first test of whether you are qualified to be a gca is to see if you can withstand the barrage of well meaning advice to not go in the profession. Sounds like you are off to a good start, but the pressure will continue!
I took the traditional US landscape architect degree path, couldn't get any summer or year long internships with architects, but did work golf maintenance and landscape construction and design (for Brickman, just sold yesterday to some big Wall St firm, but I was there when it was a small, cutesy office environment and Ted Brickman was still alive and occasionally seen around the office). To supplement my LA courses, I did a senior thesis on golf design with a friendly professor, and also took engineering classes in surveying and drainage, not to mention some business writing and marketing classes to round out my edumacation.
Feel free to contact me. At the very least, I am optimistic about your chances, knowing that everyone told me the golf architecture business was dead (in 1977 it wasn't much different than today's economy) and that I wasn't worth a damn, neither was my whole generation, but we/I muddled through somehow. I just met a young man I had similarly positively counseled and he is at least on the fringes of the biz and grateful he wasn't talked out of it.
1. Would it be worth it to go to college in the UK?
It would be a great life experience, but I don't know if you magically absorb any of the modern construction techniques there. Could go either way on that one, but I considered it, and stayed in the US because U of Illinois had a good LA program at that time. (And, because my older brother had exhausted my parents patience (and savings) with out of state college tuition paid for a drug fueled party year......)
(Side note: Really check into the programs for both golf friendly and current faculty. Professors change all the time and while its hard to know which unknown professor might touch your life, there may be a school with a glowing rep that has recently had a lot of turnover in faculty, which might make the department a mess. Also, check the emphasis in the program, as you want one with nuts and bolts, or traditional emphasis. Some programs are aimed at environmentalism, or urban planning, etc. and you might not get all that you need.)
2. In the modern business is it more important to have studied the great courses, or to have experience in the construction process?
Probably construction experience. The study of great golf courses is a lifetime endeavor, and no one would expect you to be anywhere but the start of that journey. It will help if you showed some gumption to go see at least a few of the great ones. Also, don't discount those hard to find design internships. Some guys go the construction side and never get back on the design side, as they get slotted.
Also, any good gca or landscape architect will probably know within three months if you have the overall creativity to stick in the design biz. Not pleasant to contemplate, but there are degreed landscape architects who can't design their way out of a paper bag. Despite desire, if you don't really have a design personality, you won't be a great designer.
3. What are some things that I could be doing right now to improve my chances later on?
Study up on architects you might want to work for. Nothing improves your chances in an interview than presenting yourself as someone who really knows their work and wants to work for them. Flattery will get you everywhere. If you go in to 100 offices where it would be "just a job" to you, they will sense it.
At the same time, be open minded about who might teach you the most. While my mentors (Killian and Nugent) weren't well known, they were very good at what they did and great teacher of the technical side. You might get a better experience there than with a better known firm who slots you into just one phase of a bigger production house.
4. When, if at all, should I try to do some sort of internship?
Just keep calling and knocking on doors. Most job openings in gca are a matter of timing....and persistence. I called on K and N at age 14, and kept bugging them. When I finally graduated with honors at U of I, and had done all the things they told me to do, they felt obligated to hire me. As to persistence, I might have no work this week and next week I might have five projects. I can recall hiring a guy a week after another qualified candidate came through the office, much to the first's dismay. Feast or famine, just the way it is.
So, TD is right, don't feel bad about pestering those who you think you might want to work for.
Hope this helps.