Hi Phil and Mitch,
As Mitch said, there are a lot of possibilities. Did you get any record of the work done during the restoration? Was any warranty given and/or can you talk with the "pro" to get more details?
I'd be inclined to think it's not driveshaft or u-loint related, but they should be checked anyway. The reason is that vibrations there are usually speed related, not rpm related. The driveshaft itself should be balanced. And, the sliding member at the front has to be properly aligned on the main part of the shaft. There are two arrows engraved into the two pieces to show alignment.
I'm more suspicious of the flywheel, ring gear, clutch, crankshaft, pistons and connecting rods and front extension/fan. It's unlikely to be an internal gearbox problem.
If the original metal cooling fan is being used, it requires balancing and proper alignment. Matched sets of extension and fan have 1/16" holes drilled in them for correct alignment. There is usually a single balance weight on the fan, bolted under the mounting screws. These are often lost during rebuilds. The Roadster Factory has them, but actually re-balancing a fan might be a problem. Later plastic fans did not require balancing and provide more blades for much better cooling.
If the original fan is removed and an electric fan used instead, as Mitch mentioned, at the very least the fan extension should remain in place. It serves to help dampen engine vibration. However, without the fan, I'd really prefer to see a harmonic dampener installed in place of the extension.
The crankshaft, flywheel and clutch should all have been balanced prior to assembly. If one or the other wasn't, it could lead to the type of vibrations you are experiencing. Personally I prefer to see these each balanced individually, so that any one of them can be changed out if needed, without upsetting the balance of the whole.
A loose flywheel, or even ring gear or clutch cover might cause vibrations. There are locking tab washers used on the flywheel and ring gear. If forgotten, bolts might loosen. Don't fool around with this! It can be dangerous.
It's hard to say if the chattering clutch is directly or indirectly related to the vibration, or a separate issue all together. It could indicate failed clutch disk linings, which might cause an imbalance and vibration.
Connecting rods in the engine are matched sets. Replacing one, but not the others, might lead to problems. They should be checked for trueness and balance during a rebuild. The same is true of the pistons.
The TR gearbox shift lever has a little ball and spring near the bottom, an "anti rattle" arrangement. These are often lost and omitted during rebuilds and gearboxes without them will see more vibration through the shifter. If you happen to compare a car which still has the "anti-rattle" pieces in place with a car that's missing them, you will notice a difference! This would be great news, since it's really a non-critical item.
Another relatively benign thing that causes vibrations is omitting the cast aluminum dash support bracket from these cars. I've seen many missing them. This will lead to "scuttle shake" where the dash and all vibrates. It sounds, however, like your problem is somewhere in the drive train.
Unfortunately, we can only guess based upon your description that vibration is more severe than the two above examples would produce. More severe vibration may mean more serious problems, and deeper investigation is probably needed.
You might be able to tell something by removing the starter and peering in the opening, or by removing the dust cover at the bottom of the bellhousing and looking inside with a small mirror. But, I'm afraid it would probably be better to pull the gearbox and take a really close look at the clutch parts and the flywheel/ring gear.
This can be done with the engine remaining in the car, pulling the gearbox out through the cockpit. It does mean removing at least one seat, carpeting, the center dash support and the gearbox cover. Support the rear of the engine with a jack (raise it slightly to pull the gearbox out).
Sorry to hear you are having a problem and hope it turns out to be something easily resolved!
Alan