I think you can make UK golf a lot cheaper than you'd think. And, depending on where you're coming from in the US, Scotland is easier and possibly cheaper to get to than Bandon, with cheaper lodging and no need for caddies. And while there's no disputing that even though Bandon is really special, it can't compare to Scotland.
I spent a semester at St Andrews and here's what I learned on the matter. Obviously as a 21 year old my standards were probably lower than most in pursuit of budget golf, but here's how I did it:
1) Public transit - Scotrail and busses are incredibly easy, I played 20 courses in Scotland without ever having a car. It's super cheap to get around and only remote courses like Macrihanish are inaccessible.
2) Accompanied play - This is a bit tougher but not as hard as you'd think. Accompanied guest play is dirt cheap - most I ever spent on a round with a member was 25 quid at Dornoch. If you know someone who has a membership over there, I'm sure they could help you with at least a few connections. And it could've been because I was a student, but found that people were more than happy to set me up with members to play with (it doesn't cost them anything!)
3) Lodging - Again, my standards were definitely lower, but you can always find something cheaper on Airbnb or any travel site. Guest houses are cheap and you can meet a lot of interesting people on similar trips and knowledgeable, friendly locals. My one piece of advice if you're looking to save money for the St Andrews part of the trip: stay in Guardbridge, as I know lodging in town is crazy expensive. Guardbridge is 5 mins away by a bus that runs every ten minutes, and has much cheaper Airbnb's.
4) Caddies - Playing with members tended to make this easier as you'd have someone to point you around, but one thing I found interesting is that you'd never see locals taking caddies. I played a few rounds where we just paid the full rate, for example at Muirfield and Carnoustie, where even still a yardage book would suffice. Again, I'm in my 20's and not used to the norm of taking a caddie, but I think you're perfectly fine without one.
5) Season - Golf and lodging tends to be cheaper in shoulder seasons and the weather isn't variable enough to make it risky. Some courses will require you to use fairway mats but frankly that doesn't make any difference. I was there from January-May and had a few early morning rounds cancelled due to frost and only one day where it was raining too much to play golf (for the most part rain was never worse than a blowing drizzle)
6) Courses - Scotland's gift is that it has spades of incredible courses. I'd say one's golf resume isn't complete without playing the Old, Muirfield, North Berwick (some would throw Dornoch in there too). These are worth spending money on. If you don't want to spend as much, don't bother with Turnberry, Carnoustie, Kingsbarns, etc. There are plenty of incredible courses which won't break the bank. Here are just a few I've found across the country just off the top of my head:
Highlands: Tain, Golspie, Brora, Boat of Garten
Fife: Crail, St Andrews Eden, Elie, Anstruther
East Lothian: Braid Hills, Musselburgh, Dunbar
I don't have an exact number for you, as I also mixed in hopping on Ryanair flights across Europe with friends every other weekend. But if I had to guess I likely spent no more than 1500 on golf in about 60 rounds in St Andrews and maybe 20 outside of it (most of which was from Muirfield and the 400 student annual membership at St Andrews)
I don't have any insight on Ireland or England, but am headed there in a few weeks. Hope this helps!