Yes, your overlays are lambs wool.
I will post the advice offered on the Rolls-Royce Owner's Club (US) forums from someone who knows quite a bit about doing this (Thanks Phil!):
-- First, shake the rug out to get any really loose bits of muck off. Then brush it out to get more muck off -- don't brush too gently, but don't overdo that, either. You'll probably be amazed at what comes out.
-- Then vacuum the lambswool carefully, using just the nozzle, to get lots of dirt out. You may or may not wish to put an old stocking over the nozzle of the vacuum hose so as not to swallow up any loose wool bits.
-- Then use the Woolite. Here I'd mix up a batch of Woolite in a bucket, get a sponge damp but not soaking with the Woolite, and sponge off the overlay with long swipes across the overlay in one direction, then in the other direction, then across, and so on. You may or may not have to rub somewhat vigorously in some really dirty areas. You'll probably want to do this two or three times.
I think I would avoid using a washing machine because of the age of the overlay backing. I would also avoid using a hose for the same reason. The wool is tough, but we just don't know about the backing.
-- Then you might want to vacuum again, as above.
-- If you have to repeat the process, by all means do so until you're satisfied that you've done what you can.
-- I'd recommend drying the overlay by just leaving it outside to dry. If the weather's not too hot and the sun isn't too glaring, you can put it in the sun. Otherwise, I'd let it dry in the shade, though this will take a bit longer. We find that the deck off our house is a great place to dry overlays.
-- If all else fails, then find a dry cleaner that specializes in antique fabrics and old rugs and try them. However, since the overlay is in good condition and not stained but just dirty, I think you'll have luck by following this technique.