I can't see BP or the Queen wading into a legal battle the Sussexes pick with the BBC. Not especially since all of the press releases have been so carefully worded to leave a lot of room for interpretation.
Not actively participating, just being ready to submit a statement if it becomes necessary. I would hope that H&M lawyers warned them in advance that just them saying "That's not what happened" is not going to be sufficient to back up allegations of libel.
The Queen has indirectly participated in other court proceedings involving the Royal Family: https://www.theguardian.com/uk/2002/nov/01/monarchy