A lot of wood turners use CA (Super Glue) and rub it in while it is spinning slow on the lathe (or drill press) and force it into the pores. I've done it a few times myself with good results. (be careful as you can quickly glue your hand to the wood)
On a side note: The one-and-only time I've ridden in a "TR" was when I was stationed in Key West, FL with the Navy. I was hitch-hiking home to Orlando and I caught a ride late in the evening just south of Miami with a young guy in a TR. in 1968ish it must have been like a TR4 maybe. Anyway, it was getting dark and any ride will do. It started raining so we had to put the top up. I had a full seabag in my lap. The clips that held the top down didn't work so we each had to hold it down. Rain was pouring in through the gap and down our arms. Only the passenger side wiper worked so I was the lookout for trouble. We stopped at a truck stop North of Miami and that was the end of my ride and the ONLY time I've been in a Triumph. Oh, I think it is a really cool looking car - - - - but, that one ride did me in.
When you get all the wood trim fixed, please come share it with us. I've always been very fond of real wood in the vintage cars.
Agree with John 100%. CA makes a very hard, clear and durable finish. I have used it for a number of items like yours.
A caution and a couple of suggestions:
1. Use ventilation and don't breathe much of the fumes, as they are irritating. Naturally, use gloves!
2. Use the thin version CA. It goes on much smoother and is intended for this use. At Rockler its called "Stick Fast CA Thin."
3. Pick up a can of the CA accelerator spray. A tiny squirt, and tiny is all that is needed, cures the CA instantly. You can get many coats in a short time. Sand between coats and polish as you like. It can take a very high gloss if you want to go that direction. I have been successful with auto polishing compound, as well as other stuff. You will be very surprised at how easily it polishes.