bit late in getting onto this thread but . .
I had the same when we moved into our present house . Every ceiling was artexed in a horrid comed circular patten.
Unibonded & skimmed over each room as we decorated. My advice would be to do that . Get a good plasterer in for a quote first, nothing to lose by that .
In my opinion, from your photo skimming that would not be a problem as skimmimg the ceilings is done in 2 coats ( one straight on top of the other as the first just starts to go off) if it is done properly. Ususl thing is to go over the whole ceiling with a sharp blade scraper to remove the peaks & highest spots.
But be aware as said previously some artex does contain small amounts of asbestos. I was told on a recent asbestos course that only the very early artex from late 50's has the larger quantities in it.
Most DIY er's don't know about two coat use on finish plastering
Good luck . . sure you can sort it
