I have made sanding paste using mineral oil, beeswax, and diatomaceous earth and a second batch adding some carnauba wax. I have never really used either one except as a test to see if they worked. The reason is I have always been concerned if I applied a finish over it there would be problems, i.e. mineral oil never cures and the wax. Those that use sanding paste are their “safe” film or oil finished that can be applied? Are there finishes to be avoided?
I have used Acks sanding paste and finishing polish for about five years now. The paste...I actually don't remember what it contains (they used to put a sticker on the back of their tubs listing what was in each, they no longer do), but I don't think there was any wax in the paste. The paste is one of those disintegrating grits, where the more you polish with it, the finer the grit becomes as each granule breaks down more and more.
I've ONLY ever used any kind if sanding paste, over sealed wood. Usually shellac, althought I've tried some other things as well in the past including cellulose sanding sealers as well as I think a water based sealer, and lacquer. I never use the paste directly on bare wood. It never made sense to me, and the recommendations for Acks are to seal the wood first. So my process is to sand the wood, sand it well, sand it so NO scratch marks appear (this is very important!), which can take some real care at times (depends a lot on the wood), then seal, then polish. Following this process, you can get a very nice glossy shine, or a satin shine, depends a lot on how fine you sanded and what you used to seal. Since you sealed, well, none of the sanding paste gets into the wood fibers. So even if it had wax (I also have used Yorkshire Grit, which DOES definitely include wax, and its quite different stuff from the Acks paste), you aren't embedding the paste into the wood fibers.
With Acks, I think its posible to use another finish over the remaining seal coat(s)... In fact, the recommended approach is to buff the Acks paste on starting at a low RPM, then periodically step the RPM up a bit and use a new clean piece of towel or rag, and keep doing this until you have completely cleaned all remnants of the paste off the workpiece. This is possible with Acks, I don't think its really the same with Yorkshire Grit, possibly because of the wax in it. Once you have fully polished your piece with Acks sanding paste, you should then have just the sealer on the wood. If that sealer can then take another finish, it should be possible. Shellac, which is usually what I use (although sometimes I use Mylands Cellulose Sanding Sealer), is one of those "universal over/under" things that can apparently be used over and under just about anything (except maybe over wax?) So its a prime option if you have additional finishing in mind.
In any case...personally at least, I've always avoided using pastes directly on wood. Yorkshire even, despite seeing many people use it directly on the wood...I've never really liked how it gets into the fibers. It does not seem to bring out the chatoyance the same way as when you seal the wood first. Shellac does well to bring out chatoyance. Sometimes I'll use an oil coat first, then CA (this is usually only on pens) then the sanding paste. The oil also helps bring out the chatoyance. Mylands sealer is ok, it yellows things a bit, and its more of a pee-yellow, than the warmer golden-yellow of an ultra blond shellac. It doesn't quite seem to bring out the chatoyance as much, but its still better than paste directly on the wood, IME.
I don't really use water based finishes, not really a fan...they all seem to have very high solids content, and sit more as a film on top of the wood and don't seem to penetrate and bring out the brilliance of the wood, so I don't have any experience there with regards to how well polishing pastes work on them. I did try Craft Coat for a while, which I didn't realize at the time was a water-based finish. Initially I thought it looked fairly good, shiny, but it actually shrinks and forms a hard film coat over the wood. Since it shrinks, it takes on grain characteristics underneath, and that actually made it fairly hard to polish well. Sanding it was VERY hard, it doesn't sand well. This was my longest used water based finish, and one of the reasons I kind of wrote them off for good some time ago. In any case, if you use water based finishes, I would make sure they are compatible with a sanding paste, if you use any as a seal coat.
Oh, and for sanding pastes...I don't remember what Acks has in it, but I don't think it has any oil or wax. Its a brown colored gritty paste, but at normal room temperatures (i.e. ~70F/21C) its relatively soft. Its a decomposing grit, so it starts out feeling coarse (they say its about like 300-600 grit), but as you polish it decomposes and eventually becomes an ultra fine grit, which can be as high as 1500 or higher. If I am making something I want to use Acks as a finish, I sand to at least 600, sometimes 800 or even 1200, depends on the wood. Despite the grit feeling coarse at first, it very rapidly breaks down, and does not leave any large scale scratches...but it definitely polishes up things very nicely. With the right woods and pre-sanding, you can even get some fairly glossy finishes, although most of the time its a satin sheen, and with shellac as a sealer (I usually use the Zinssers Bullseye Seal Coat) the chatoyance is very good to extremely good.
Such is my experience with sanding pastes. I say, seal the wood first, keep the paste from saturating the wood fibers, and you'll probably have a better chance at using another finish over top (although, any finish that needs penetration, say oil, obviously wouldn't really work that way.) If you need to use a finish that penetrates, you could try applying that first, then sealing, then use the sanding paste.