Some time ago, I dropped on a proper eBay bargain, and snagged 18 5" castor wheels for a measly 20 squids delivered:

A few more squids were handed over to Bearing Station in return for a set of axial needle roller bearings; and the lathe was fired up for a bit of turning malarkey. The results are now in:

The red plate had a 25mm hole bored in it, followed by a 42x1.5mm recess for the bearing. The "spigot" was turned from a remnant of steel bar I had left over, and was turn to be a close sliding fit through the plate & bearing, and also through the centre hole of the castor plate. Drilled & tapped for an M6 bolt, which, in conjunction with a big washer, is used to hold the castor in place. It's not particularly tight, just tight enough to remove any slop.
Assembled:

The castor now swivels wonderfully

The only problem is, because the axis of rotation is directly above the centre of the wheel, it's not self-steering.... No problem, Mk2 will be 2 plates sandwiching the bearing, with the castor bolted on at an offset, so it will steer exactly like a shopping trolley (i.e. anywhere but where you want it to go!

)