As colliewoman says, I've done it.
My own Jersey's milk, handmilked, straight into a bottle and into the lambs. They absolutely thrived on it, much better than on the ewe milk replacer in my opinion. And absolutely no scours - which frankly surprised me, but there wasn't any, which is less than I get on the ewe milk replacer.
I had to feed more quantity than I would of the ewe milk replacer - 1.5L per day per lamb as against 1L of the ewe milk replacer.
BUT - yes, of course there's a BUT

- I did find that if I took an orphan that had been on cow's milk and set it on a ewe,
then it got scours, and real bad scours too - I lost two lambs and had a couple more really quite ill before their systems adjusted. So now I feed them half-and-half cow's milk and ewe milk replacer until they are past the possible-to-foster stage, after which they can go onto straight cow's milk if I want.
BH once tried one lamb on shop full-cream milk. The lamb died. I am pretty sure it needs to be untreated milk straight from the cow. Certainly unhomogenised and probably best if it's unpasteurised too.
There are quite a few farms around these parts who keep one or a couple of Jerseys for milk for the pet lambs, and everyone I've asked says they don't have any problem with lambs scouring.
I haven't tried it with milk from other than a Jersey, nor from a cow from a different farm. So we're talking high butterfat, high protein, full of local antibodies. Oh, and my Jersey's milk does not upset people who are allergic to cows' milk; apparently some lines of Jerseys don't have the mutation that causes the allergy. Whether or not that is relevant in terms of how their milk suits lambs I don't know - but I would certainly advise proceeding with some caution if using milk from a different breed, just in case.
Note that I would always ensure every lamb gets ewe colostrum or ewe colostrum replacer; I would never rely on cow colostrum alone, although I will give second and subsequent feeds of cow colostrum once a lamb has had a first good feed of ewe colostrum or ewe colostrum replacer.
I haven't heard of injecting cows with sheep vaccines and can't envisage myself doing that to any of my precious Jerseys

, probably in any case but certainly not unless it is approved by the vet.
Oh, and the lambs on Jersey milk and on half-and-half have all finished fine and reached good grades (good conformation is not expected of pet lambs generally.) I do think we've had less losses and less problems when they've been getting milk from the farm Jersey - it's a magical fluid, in my opinion! We've even kept three ewe lambs reared thus on for breeders and tupped two of them as hoggs - to a Shetland, mind, so just wee lambs, we are hoping. (One's scanned with twins

, the other with a single

)