Karl. The trip meters each drive via worm gears from a worm on the input shaft of the speedo. The gears are plastic and can and do crack so they do not hold onto their respective shafts, resulting in jamming and teeth shearing from the gear and hence loss of drive to the trip meter. Additionally should the mechanism get fouled with oil and jammed, similar result occurs with sheared teeth on the gear.
Replacement of these gears is a common task for reputable service people and is best attended to by them. DIY involves stripping down the instrument, removing the offending gear, sourcing a replacement and putting the instrument back together.
I have done repairs similar to this on the Smiths chronometric speedos where I have made the gears (brass and phenolic board) and replaced them. More involved than the magnetic speedos of the TRs but I would prefer to outsource the work these days given a choice. Modern replacement gears are (I beleive) produced from a higher quality plastic also.
Should you want to check for interest or peace of mind, disconnect the cable, undo the two retaining nuts and remove the speedo from the dash.
Dismantling the speedo is straight forward. The bezel has tangs on the rear that disengage with the body by turning a few degrees, releasing the bezel and glass.
Don't touch the face as the paimt is easily damaged. Two screws on the rear of the body release the inner mechanism from the body, at which point you will be able to see all the mechanism.
Would love to know the outcome.
Lionel