I would be amazed if a new chuck or a backplate with a 1 1/8 x 12 tpi thread was commercially available.
You are looking for spares for a machine that is nearly 80 years old!.
The ML4 went out of production in about 1944. And was superceded, post war by the ML7 in 1947.
The best that you will get are either secondhand parts (which may be worn or damaged ) or modern parts that can be modified to fit.
For such a machine there is no "Make or Buy" choice, apart from any used components.
"Make" is about the only option.
The last ML4 with which I had contact, about three years ago, needed a gear cover. It was made out of sheet Aluminium made alongside it, I managed to modify a gear for a ML7 to be used on it!
To make fitting a 4 jaw chuck to it (A n even older machine with the original 7/8 x 9 BSW Mandrel, so possibly 90 years old ) I made an fitting to go on the Mandrel so that a "modern" Myford chuck could be fitted.
Comparatively speaking, only needing to sleeve such a chuck, you are lucky!
Making such parts will be a useful learning exercise for you, and familiarise you with both the machine and machining processes. As would making accessories such a Centre Height Gauge, to aid setting tools in the toolpost. So would making a mandrel Handle, for when you start using Taps and Dies to cut threads.
If you have no books on machining, may I suggest buying and studying:
L H Sparey "The Amateur's Lathe" Basic, principles, leaning towards the ML7 in the illustrations, but still useful.
Ian Bradley "The Amateur's Workshop" Basic principles, of Fitting as well as lathe work
Tubal Cain "the Model Engineer's Handbook". An extremely useful reference book, with data on all sorts of aspects that you will encounter as a machinist.
You will find a set of Zeus Charts invaluable. I constantly still use mine, dating from 1958.
As you find a need for them, you can buy some of the books in the "Workshop Practice Series". Various ones deal with things such as Tool Grinding, or Taps and Dies; and many other things with which a Model Engineer may come in contact at some time.. You are unlikely to need all of them.
Or you could also buy Harold Hall's book "Lathework" It will deal with more modern machines than yours, but the principles remain the same, even if the detail differs.
A while ago, Neil Wyatt ran a short series in Model Engineers Workshop, on Lathework. The machine shown there was a modern machine, a Sieg SC4, much more sophisticated than a ML4 but obeying many of the same rules.
I have had these books, for years, and refer to them from time to time, some more frequently than others, but ALL useful info.
Howard