As a dedicated fan of the big '50s neck shape I'd say please don't do it!! :(
It's a fairly major piece of surgery - removing the finish, reshaping the neck then refinishing it; could be difficult matching the colour of the existing finish. If it was a maple neck, you could maybe leave it unfinished (Zakk style) or wax it, but mahogany's a lot more porous and I'd guess it's unwise to leave it unfinished.
In theory, the thinner neck will be less stiff and therefore may not stay in tune as well, but that's no more of a problem than with any existing slim-taper neck. And of course it'll devalue the guitar, a lot.
I have no idea what it'd cost, but it can't be cheap. I'll be interested to see what Feline and Wez have to say.
Unless it's a really spectacular bargain price, I'd have thought you'd be better waiting until something similar with a '60s profile turns up.