Haven't been to BA's Mumbai lounge yet, as it's only open when there are BA operated flights going. I presume you are going on a BA operated flight?
But I have heard that there are at least 2 showers, so you should be in luck.
However like all lounges in Mumbai airport don't expect too much, I've been in the Clipper lounge and Lufthansa's lounge, both were ok, but nothing to write home about, still better than sitting in the terminal.
Mods, can this thread be moved to "Your questions"?