Well, the answer to the question is complex in some ways and can get political (as much as watchmaking does anyway), but generally yes, it happens and I've seen it. It's not common in my experience, but as noted it will likely be more common in the future as more and more companies, including Swatch, cut back on the availability of spare parts. Some things are obvious, and some not. For example a fake bezel insert is usually pretty easy to spot, but a fake sapphire crystal may be more difficult, and "fake" movement parts would be even more difficult in some cases.
For the movement parts you would have to define what you mean by fake though. Many parts are interchangeable between branded and non-branded movements. It's a common belief that the Omega Cal. 1120 and the ETA 2892A2 are "the same movement" which is not true at all, as
many parts are not compatible between these movements, but some are. The same thing is true for some 7750 based movements that Omega uses, like the 1151 in the OP's example. Back to the 1120/2892, and for example I can't use a rotor bearing from a 2892 in an 1120 - they are completely different designs. I can't stick a barrel bridge from an 1120 on a 2892 - the bridge for the 1120 is quite a bit thicker and will jam the rotor - found this out when I tried to replace a damaged barrel bridge on an IWC I was servicing using a spare Omega bridge I had in stock - no dice.
But if I use an ETA branded intermediate and third wheel (common wear parts in an 1120) they will work just fine, so are they fake because they don't come in an Omega package? What about an ETA 2892 mainspring compared to an Omega 1120 mainspring? Both are made by Nivarox. So personally as long as the quality is the same, I will use interchangeable parts, so for example when someone sends me a manual wind Panerai with an OP-whatever movement inside (really an ETA 6497-2), I use Omega Cal. 2201 replacement parts in it, because they are just as good, if not better, quality than the original parts in the watch.
Aftermarket parts are nothing new - the entire Bestfit system that many in North America are familiar with are essentially an aftermarket parts manufacturing and distribution company. They were known for producing high quality items that were as good as the originals. There are other similar system that have been around, so this has gone on for a long time, and was probably more common in the past as watchmakers would have large cabinets of parts, including movement parts, crowns, hands, etc in the Bestfit system, so rather than order a part from one company and another part from another company, they had compatible spares on hand for a broad selection of movements. I still use those parts for repairing defunct brands or brands that have changed hands and vintage parts are no longer available from the manufacturers. Of course I have a parts account with Omega, so I buy Omega parts directly...
Now Rolex is once again cutting off more and more independent watchmakers, and there is a big push to have someone step in with aftermarket parts. I have had the experience of ordering Rolex parts from my local supply house, and getting aftermarket instead of genuine parts that I asked for, and they looked rougher than the worn parts I was replacing, so I sent them right back. I was able to source what I needed from someone else who had genuine parts, and I actually just stocked up on a bunch of Rolex parts from a watchmaker I know who is retiring and selling off some of his stock.
Based on some very loud voices I hear in the watchmaking community, many watchmakers who have who have had their parts accounts closed by no fault of their own, there is a willingness to use aftermarket parts. Many watchmakers feel they have been screwed over by companies after jumping through all the hoops and demands made on them, and then having their accounts pulled. Many of these people have no qualms about using aftermarket parts.
I would suggest if you have concerns about your own watchmaker using aftermarket parts, just ask them about it.
Cheers, Al