At first I thought the forks could do with some extra oil for damping, so I topped them up. This seemed to make the problem worse, with very harsh front suspension, and oil running down the forks onto the tyre. Not a good idea!
Today, I decided to remove and strip down the forks, and it was immediately clear what the problem was - a broken spring! All the weight at the front of the bike was taken up on the one 'good' spring, which proved just too much for it - hence the tendency to bottom out!
I've now ordered a pair of springs from NSA Classic Motorcycles of Worlingham, Beccles, Norfolk, so Little Willie should soon be back on the road again. I'll document the rebuild and add it to this website as soon as I get the springs.