G'day Gordy,
Well done finding a HW in .45 Eley. Also known as the .450 3 1/4" BPE (black powder express). We see them here in Oz occasionally, I've owned a couple and currently hunt with one. What sight is it fitted with, should be some form of folding leaf. Those thin rim cases that ssdave has offered you should work fine but I would neck anneal them first as they are old brass. I've also come across HWs in .45 Eley that accept thick rim cases rather than the thin rim version. You can use the new Hornady .450 Nitro Express cases with the thick rim in these rifles. It all depends on what reamer was used with your rifle  I suppose the thing to do would be to try a Hornady case first and see if the breech block closes easily. 
You can use .450 NE dies to reload and as this was an express rifle and was designed to shoot flat to about 150 yds with a 260 to 365 gr bullet (1 in 36" twist). The Ideal 457122 HP (330 gr) works well in mine. I use BP loads or Trailboss. 
Here's mine, I'd be interested to see a photo of yours.
Spud

I didn't notice that when I first looked at those photos of the cases (old eyes!). I reckon they probably were Berdan primed that someone has drilled out to take another form of primer. I don't think early Eley cases were ever made for Boxer primers.
Spud

Just a thought could the rim recess just be cut deeper for the newer style of brass or I am missing something? I do understand that it might detract from its originality and also possibly some of its collector worth.

The cases probably looked like this before someone drilled out the Berdan primer pockets.

In the dark past I drilled out Berdan 43 Spanish cases to use shotgun primers. Worked fine with black powder.
Cheers,
laurie

Gordy,
I went checking thru my records to see what cartridge cases I used in my HWs in .45 Eley. Originally used Bell .450 Nitro Express 3 1/4" brass. These had a thin rim approx. .040" wide and fitted my first HW. The next two HWs both preferred a thick rim case. Initially I used the Bell case with a wrap of thin copper wire ahead of the rim as a spacer. This worked but was not ideal. Then Bruce Bertram started making .450 NE cases in both the thick & thin versions so that solved that problem (Waterman is 100% correct so they were available in the USA, I don't think BB is making the thick rim these days). Finally Hornady .450 NE brass appeared (great brass) and I've been using those.
If you want to learn about reloading for the .450 BPE suggest you track down the following excellent references...
"Shooting the British Double Rifle" 3rd Edition by Graeme Wright and...
"Finding out for myself - Part XV" an article in The Double Gun Journal - Autumn 2008, Vol. 19 Issue 3.

Just make sure that HW is in good shape & the action is tight before you start shooting it. Many of those rifles saw hard use in Africa and India and were not looked after. And as Old Win points out not that many were made and they are highly collectable Winchesters.
They do make a great hunting rifle...so enjoy!
Spud

Here is my 45 Eley