As others have indicated, formal 'will readings' don't take place and (other than in the works of Conan Doyle, Agatha Christie, etc) probably never did.
The executor(s) MAY immediately tell beneficiaries (or, indeed, anyone else if they so wish) the contents of the will but they're not obliged to do so. However once probate has been granted, the will becomes a public document, available for anyone to consult.
The probate process can be fairly quick if the will is simple and the testator's affairs were uncomplicated. (I'd completed probate of my father's will, and 'tied up all loose ends' only 2 or 3 weeks after his death). However some complicated wills/estates can take years to sort out (especially where, probably unwisely, the executors have decided to use solicitors to sort out the will).
Chris