Matt:
Re: We can machine them here for you, all I need is the base and the height that you want. If you have a set of old rings that makes it even easier. If you don't have rings I need the diameter of the action where the base is attached.

Thank you for the kind offer, but see below, plus the fact that I do not have a base. That is the problem, plus the problem of trying to determine the diameter of the #1 MK III or Long Lee actions left side-wall, behind the front receiver ring, as there is no way I am able to see how to obtain a radius measurement.

Eagle27:
Re: I had a set of the original style Weaver rings in 26mm in which I had a 26mm Kahles scope mounted.

I'm not referring to the top mount rings; I have a number of sets of them & they are perfect for mounting a Kahles Helia Super 2S2 2.5x20 scope on a Mauser, but they do not work on a #1 Mk. III, except when using the now unotainable ATI no drill scope mount, which resulted in a very highly-mounted scope. I am referring to the Weaver side-mount system:

http://www.weaveroptics.com/rings_bases/bases/side_mounts/

https://www.midwayusa.com/product/1009209933/

Docbill:
Burris makes a ring with a plastic insert that I have opened up to 26mm. Maybe not period correct but a workable solution.

I guess their Rimfire Signature rings would work on a Weaver TO-1 mount, but as you said it is not period correct.