Nestled inside the front hubs on cars with wire wheels is a grease cap. The grease cap has a 1/2" long 5/16"-24 bolt sticking through a hole to attach an extractor to. (The head of the bolt is welded to the inside face of the grease cap.) The extractor, Austin part number 1B4339, was supplied in the tool kit of the 100, 100-6, and early 3000 (through 10610 BN7, 10564 BT7) cars with wire wheels. (The Austin-Healey 3000 Service Parts List, AKD 1151, 5th edition)
If you don't have the factory tool, consider making one of the following versions:
The first tool is made from:
You want a soft bolt (Grade 2) as we're going to drill and tap a hole to screw the grease cap screw to. At the threaded end of the bolt, drill a 17/64" diameter hole, 3/4" long, along the axis of the bolt. Tap the 5/16"-24 threads.
For the washer, you can use anything that will span the hub.
To use, thread the nut all the way up under the head of the bolt, put the bolt through the large washer, and thread the bolt on to the grease cap screw. Thread the nut back down to the large washer. Continue to turn the nut against the washer using a wrench til the grease cap pops free.
To make the second tool, you need:
On one end of the all-thread, machine or file a couple of flats 180 degrees apart to put a wrench on. Chase the threads around the flats with the nut. Permanently attach the coupler nut to the other end of the all-thread, screwing the coupler nut about half way on. I used Loctite 272 retaining compound to lock the pieces together, one could also drill and pin them together, or braze them.
Usage is similar to the first version of the extractor. Install the coupler nut on to the grease cap, slip the large washer on, and install and run the 5/16" nut down. Continue to tighten the nut against the washer with a wrench until the grease cap is free. Use a small adjustable wrench on the flats of the all-thread to keep the all-thread from rotating, if required.
The second version is probably easier to make if you can find a fine threaded coupler nut. I ended up making a coupler from a 0.9" long piece of 1/2" hex stock. The design of the second version is also closer to the extractor supplied with the car, but I don't know what the dimensions of the original design are.
Another thought is to make an adapter that attaches to a slide hammer, but I'll leave that as an exercise to the reader.