Wobbly headset, maybe...

might be the front brake that wobbles a bit?

good idea!

check the wheel bearings nic
had this issue once, was so sure it was the headset messed up a nice one trying to tighten it and also a stem
turned out the roller bearings needed renewing in the hub

I'll try a different wheel, but having badly serviced a hub before, I know what a dodgy wheel feels like!