I would - respectfully - point out that's not the situation.
It is (was?) a promise in the sense that the government is there to enact the will of the people. The majority of people who voted, voted to leave (personally I was in the minority).
So now (in engineering terms) it's pretty simple. If everyone sits on their hands, the UK will leave the UK as scheduled in March (and so fulfil the wish of the people).
The actuality is the unedifying spectacle of lots of shuffling, muddying of the waters and lining of pockets as anyone with skin in the game scrabbles for whatever they feel is to their own advantage.