They're both solvents with slow 'flash' times (meaning they evaporate slowly). Blush eraser often comes in spray cans, although you can also buy it bulk - for mixing with paint, to slow curing time; it's purpose primarily is to allow any moisture trapped in the finish to percolate to the surface before the finish cures.
So essentially you'd spray the offending areas with blush eraser, which would 'melt' the lacquer. It'll re-flow, and when it dries the cracks will be gone. It may take several applications, and you'll have some work to do afterward in terms of sanding and polishing.
Butyl cellosolve works the same way but it generally comes only in bulk (AFAIK). I use it when there's a single crack or defect I want to address, I take a pipette and apply it to the area directly. It has a VERY long evaporation time, not very volatile at all, and it's not quite as aggressive in terms of how quickly it dissolves lacquer.
I think blush eraser is what I'd do here, but a little bit goes a LONG way and I'd practice spraying it on scrap first so you get a feel for how it works. And of course, you don't want it on anything you care about, it's an equal-opportunity solvent.
AdmiralB nailed it. I bought Cellusolve in a smallish container, long ago though. (pint I think) You need to keep the surface flat while the solvent melts the lacquer... or it will run. I used it last year to repair a 70 YO guitar in one area. I just brushed it on quickly. I bought it over 10 years ago I think. There used to be a video of someone remelting the lacquer on a vintage Martin. By using the Cellusolve they kept the decal intact. I thought it was done by Gruhn, but I haven't been able to find that video in years. http://www.hvchemical.com/butyl-cel...ueNAy2KH6HPDlqCqOoEC1JDY9HHJwE4EsgaAp_c8P8HAQ
I would think finishing (especially woodworking finishing) supply houses should have it. You might contact Touchstone Tonewoods (touchstonetonewoods.co.uk), they're a luthier supply shop in the UK. They show some finishing stuff; not blush eraser, but I'd bet they would have some ideas as to where it might be found.
It's not just used by musical instrument finishers, though - people doing furniture restoration would use it, the first video I saw on its use was from a furniture refinisher.