Someone correct me if I'm wrong (IANAL) - but in the UK it is "loser pays" - ie Tesla will be liable for the BBC costs.
See Samsung vs Apple, initially Apple didn't have to pay Samsung's fees, but after Apple annoyed the judge with their non-apology apology, the judge ordered that Apple pay [1]:
As to the costs (lawyers' fees) to be awarded against Apple, we concluded that they should be on an indemnity basis. Such a basis (which is higher than the normal, "standard" basis) can be awarded as a mark of the court's disapproval of a party's conduct, particularly in relation to its respect for an order of the court. Apple's conduct warranted such an order.