If they TRULY regretted it, they would replace their Ambassador.
Its not the point. The ambassador can say whatever he likes. After all, it was never for public consumption. HMG don’t have to accept or believe his comments, but he is there as their “inside man” and they really have to back him up. The fault lies with whoever has leaked these messages. That has several affects. Firstly obviously its very embarrassing for the Ambassador and HMG; but secondly if ambassadors get the idea that they cannot speak freely because their comments will get leaked, they are not going to express themselves freely, particularly if the precedent is established of HMG throwing them under the bus.
I think personally his position is now untenable, but because Trump tweeted his displeasure to all and sundry, they can’t recall him. My guess is that he will be “asked” to resign. Apparently he was due to retire at the end of the year anyway.