Not GiveMeSpanokopita, but no leak from anyone was necessary for this story to get out.
The court’s file is considered to be a public document (not the family courts, though) and the court openly lists what claims have been filed in its various offices each day. Anyone can check it. The newspapers pay a journalist to check it every day to see if there’s anything juicy posted, and then if its something they want to cover they will make their own enquiries.
So some time after the claim was filed on 5 March it would have been listed on the court’s website (it’s not done in real time, it normally takes a few days to appear). The Times would have seen it, and seeing the defendant was Sentabale, and this might involve Prince Harry, would have contacted both parties for a statement. Doubtless sent someone down to search the court’s file for this case too, to see if they could see anything!