Mechanism for Brown Dwarf Fusion
Your scenario is, of course, rather speculative and considers the very distant future. I don't like that article you're referencing, some stuff seems inaccurate (for example the fraction of baryons locked into stars is tiny and will always remain tiny).

If two brown dwarves collide (extremely unlikely, but if you wait long enough, anything with some tiny non-zero probability eventually will happen), they form a new star with their combined mass. Since the brown dwarves have not undergone any hydrogen burning, the new star has the same composition as a new-born star of the same mass would have had (except for lithium) and, if massive enough, will evolve just like an ordinary M dwarf (or so). In other words, it will fuse hydrogen into helium as all stars on the main sequence.