Are American English and British English growing closer together or drifting further apart?

I could find no studies looking at the closeness of British and American dialects of English. But I would say that the question is formulated too nebulously to make it possible to answer easily.

There are studies that show dialect leveling (or reduction in regional variation) within British and American English. But this is always illustrated on specific features and harder to specify in aggregate.

Also, while there seems to be less dialectal variety within broader 'English' regional varieties, this has been ascribed to geographic mobility and not necessarily to media influences. While there are fewer Northern or Southern Englishes, there are still huge differences between the varieties that remain.

I don't think this at all translates into leveling between British and American English. There is virtually no regional mobility (to have an impact) and the influence of global media is incredibly shallow - just observe the struggles of transplants in either direction. If media had the sort of influence everyone imagines, you would expect a lot more of merging. There are a lot of examples of mutual impact but none of them systemic.

I would propose that both American and British English have been and still are developing more or less independently and will continue to diverge over time. I would suggest that literacy and media provide more of an internal cohesion for each variety than a vehicle for mutual comprehension or a future merging.