Andy - you've a number of options here to fix all this but also a big warning. The original Artex product contained asbestos (the new stuff doesn't) so you need to consider this if you decide to remove the existing ceiling and take appropriate precautions - also Joe is correct as to why people originally Artexed. You might be lucky to have a ceiling in decent condition (Artexed as a style choice - yuk!) so skimming is a solution
First, if you have access to the room above lift some floorboards to see what type of ceiling you have, lath & plaster, fibre board (common repair after bomb damage in WW2), etc.
Assume you'll take the ceiling down -
ceiling rose - remove the rose by cutting around with plasterboard (jab saw) and lower from ceiling. You may have to check for anchors from above by lift some floorboards. You could leave the rose in place and anchor/reinforce as described below for the cornicing.
cornicing - vacuum all crap & dust from above (again remove floorboards), lay jute scrim along the top of the cornicing, dampen both, apply a good layer of plaster (we use plaster of Paris 'cos it dries very quickly) to the scrim/cornice/wall/joist interface. This holds the cornicing in place when you drop the ceiling and prevents cracking.
Cut around the ceiling, and through the laths, at the cornice/ceiling field interface, cut around the rose. Pull down the knackered ceiling; tip - seal doors with tape, wear decent mask/goggles 'cos the dust'll be awful.
Board the whole ceiling (fix with screws), skim. What you'll end up with is a brand new ceiling with terrific original rose & original cornicing.
A Victorian house = 100 years+ of paint on the rose & cornicing resulting in lack of detail. Problem with restoring (paint stripping) the rose will mean you'll have to do the same to the cornicing - it'll take you weeks to do it! What you'll be left with is something that looks newish and maybe out of character with the old house, so you might just want to leave to old paint in place. If you want something that looks new just rip the whole lot down and stick up some foam stuff.