Financial institutions are likely to start putting contingency plans into action in March.
But Howard Davies warned that "some" jobs at the bank will go and that the damage of leaving the EU on the City will be "quite considerable over time" regardless of the outcome of negotiations, the timings of which are becoming "very tight indeed".
And, without more detail on a transitional period, the City's financial institutions are likely to start putting contingency plans into action in March next year he believes.
Speaking on Sky News, Davies said business is more concerned that "it's taking quite a long time to get to the nitty gritty of what a new trading relationship with the EU would be".
"Certainly some time has been wasted up until now in negotiations which haven't really got anywhere," he said.
However, the publication of several position papers over the summer and meetings with business "of a more intense kind than before" over the summer signalled that the government appreciated the urgency of getting to "proper negotiations".
"I think Theresa May's Florence speech was a significant step forward," he said. "Whether that will be a big enough step to unblock the negotiations we'll only see in the next few weeks."
Asked if he thought the government realised the damage being done to the banking sector by Brexit, Davies responded saying: "I hope so, because we keep telling them."
"I think the financial sector has been telling them very very firmly. Now some of this is inevitable. I'm afraid if we go in for Brexit we will find that jobs will leave the city, and that there will be a re-balancing of financial activity within Europe.
"The City has been astonishingly dominant on European financial markets in recent years. In our case it will be relatively small because we're not a very large player in European financial markets," he said, with some lending and forex trading activities in the region continuing via a subsidiary in Amsterdam.
The damage to the City will be "quite considerable over time" he said.
"Up until now people have centred their European activities, and we're talking about particularly the American banks, the Japanese banks, even the Chinese banks, and they have chosen to put the enormous lion's share of their activity in Europe based in London. And they are now re-balancing. That's going to happen whatever the outcome of Brexit negotiations are."
He continued: "So the game is can we provide an outcome which minimises that cost. It will not remove it. It will definitely be a cost. Can we negotiate enough market access so that cost is at the margin... is a few thousand people, not tens of thousands."
He added that a "no deal" Brexit would be "bad news" and result in quite large moves of people would take place.
Davies also warned that without detail of the transition period by the first quarter of next year, banks would accelerate moving people out of London.
"The Bank of England has asked every financial institutions to prepare a contingency plan in the event of a 'hard Brexit'... and if nothing is certain by the end of that first quarter, by March say, then I think they will trigger those contingency plans because they won't have time to get them in place by March 2019 unless they start at least a year ahead. That's the moment we need to know what the transition arrangements are."
"It's very very tight now. Very tight indeed," he admitted of the timings.
"What we are seeing now is the un-wisdom of triggering article 50 when we did without knowing what the end point was. And I'm not really sure why we did that, because that was purely domestic political reasons, but it didn't start negotiations off in a good way."
He also said there were those in Europe hoping that the more the negotiations are dragged out the more businesses will move to cities across the region. "They have no incentive to get on with this and that's the real problem," he said.