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.

When limited to supermarket cheeses, I would try 1/3 mozzarella and 2/3 muenster. Braciole is served hot, so mozzarella would have lovely Italian flavor but would liquify too much, while muenster would have the right texture. The pork would not have to cook as long as beef, but you run the risk of cheese seeping out into the sauce rather than remaining as an intact layer of the meat roll.

If the only issue were flavor, an aged provolone takes on a similar characteristic of other italian hard cheeses such as parmesean, and so I'd have replaced the provolone with a dry mozerella augmented with extra parmesean, or pecorino romano. (and at least in the U.S., dry mozerella is what tends to be sold, not the fresh stuff)

As we're dealing with melting characteristics as well, as Kathi's pointed out, mozerella's just going to melt wrong. I'd likely go with fontina, a young asiago, or manchego (a young one, not manchego viejo).

To emulate a younger provolone, use less of the aged cheese and use a better melter, like fontina, monterey jack or even a mild swiss.