Best way to restore my old vintage wooden cricket bat
This is what I would do if I had to do this:


Use fine sandpaper to remove the stickers and the lacquer.* You may want to sand down the scratches as well, but I can’t tell by the photos if that would be a good idea, or that they are too deep (they seem to be so) and trying to fix them would deform the shape of the bat itself.
Fill the large crack with wood adhesive. Keep in mind that some adhesives (like PVA) shrink as they dry. If the crack is too wide it may be a better idea to mix some fine sawdust with the adhesive instead of using plane adhesive.
Measure and draw a pencil line indicating the desired shape of the bottom part. Use rough sandpaper to shape it and then polish it with fine sandpaper.
Put finish on it. (From the little search I did on what finish to use for the purpose, people advise against using hard finish, as it may end up cracking. Polyurethane spray was a suggestion I found reasonable.)


*You asked if sanding it down will cause the bat to lose its originality. I’m not sure what you mean by this - what you want to do:  remove the stickers, the cracks and deformities, will cause it to lose its originality in a way. If you are concerned about deforming it somehow – then no, the finest grade sandpaper is used for polishing and it won’t deform it, or leave scratches.