What is the best substitute for Provolone?
There are two main types of provolone: piccante (sharp) and dolce (sweet).  Most braciola recipes I've seen call for the sharp version.  Sharp provolone tastes very similar to cheddar, which I assume is abundant in the UK.  The issue is that "real" cheddar is a hard cheese and provolone is only semi-hard.  Here in the US, we have a softer version of sharp cheddar-style cheese that is made in Wisconsin.  If a similar semi-hard sharp cheddar variant is available in the UK, I would recommend that.

It should also be noted that the term "braciola" (or sometimes "braciole") means a lot of different things to different people.  To most Americans, likely including the author of your book, it takes on the Sicilian meaning of an involtino.  Sometimes the meat will be filled with a soft cheese mixture like mozzarella mixed with grated pecorino romano.  However, if the recipe calls for a harder cheese like provolone, it is likely referring to a variant that is very popular in the US in which the provolone is cut into thin slices (usually using a deli slicer) and layered on the meat before the meat is rolled.  Therefore, any substitute for the provolone should ideally be thinly sliced.