I've now solved my own problem and hopefully the solution might be useful to somebody else. With hindsight it seems obvious that a corrupted Spotlight index on my Time Machine volume (not my boot volume) was causing the problem. What distracted me from realizing this is that indexing of the volume was turned off using Spotlight's privacy settings. Regardless of this, Time Machine needs an index of its destination volume and goes ahead and creates one anyway. The corrupted index (and I've no idea how that happened) was extending every run of Time Machine by several hours so the whole thing seemed to run continuously. Among the (many!) things that didn't solve the problem were: Rebuilding the Spotlight index on my boot volume and repairing (with Disk Utility) my boot volume and Time Machine volume.
To correct the index corruption I just needed to delete the old index using the terminal. The command to do this is potentially very dangerous so make sure you check it very carefully before you hit return. Type the following into a terminal window:
cd '/Volumes/Time Machine'
pwd
sudo rm -rf .Spotlight-V100
If your volume is called something other than Time Machine then replace its name on the first line. The second line is just to make sure you're really in the right place before typing the dangerous command on the third line. Type it very carefully or cut and paste from here! After you type the sudo line you'll be asked for your password to confirm that you want to delete the index that "belongs" to the system. If you're not using an admin account you'll first need need to type a line consisting of su, a space, and the name of an admin user. You'll need to type the admin user's password once for the su command and again at the sudo command.
The delete will take a while to run and your next Time Machine run will stick on "indexing" for hours while the index is rebuilt. Hopefully, like me you'll find that Time Machine runs quickly after that.
