My experience is this: I booked several award seats using Virgin miles, plus a couple of biz cash tickets on KLM and AF. I added all the booking codes to my KLM profile. I then started a web chat with someone at KLM, and asked him to manually add the AZ number. At first he couldn't do it. Then when he succeeded, I rolled off all the other booking codes. In future, I would not add the bookings to my KLM profile, as the chap said '...there are already two FF numbers (KLM and Virgin) on many of these', which I asked him to delete.
So far: two segments on AF, the FF number appeared but with 'null' and did not show Sky Priority. I managed to get two new paper boarding passes at Newcastle, showing Elite Plus and Sky Priority. For KLM, I thought I would have to do the same, but I've managed to download three boarding passes to my phone and they all correctly say Elite Plus / Sky Priority.
On all segments so far I haven't been able to choose upgraded seats complimentarily, although I'm not sure this is a SkyTeam benefit. The SkyPriority is very useful at CDG, plus at NCL at least, the AspirePlus side of the lounge. Having been BA Gold for 10+ years, then Star Gold for a few (still am the latter) I am hoping a combination of maintaining Sky Elite and Star Gold will work better than putting everything into BA/OW. Plus I've a pile of Avios to get through, and flying all biz with BA is nearly the same as BA Silver anyway.
Lounges at CDG and AMS are good as well. Getting back on point: boarding passes have worked in all lounges so far. The only thing I can't do is add the ITA card to my Apple Wallet. It just won't go in, but still shows in the ITA app.