"Whenever there is any update to Brexit negotiations or parliamentary votes, we see a spike in support requests as customers try to understand what it all means for the pound and our Brexit Support Desk will be anticipating lots of queries today and tomorrow once the results are known".
Mr Cox's advice was issued the morning after Mrs May's dash to Strasbourg to finalise a deal with European Commission president Jean-Claude Juncker which she said would deliver "legally-binding" reassurances for MPs to ensure the Irish backstop can not be permanent.
However, this concession from the European Union failed to impress parliament as the new type of backstop had no clear end date and can not be ended unilaterally by the parties.
The stinging 149-vote defeat stripped away May's control over the course of Brexit and handed it to Parliament, which is divided about what to do next.
"We won't know how long that extension will be, that's for them to decide".
However, he also warned that "the legal risk remains unchanged" that Britain would have no legal means of ending the controversial backstop without the European Union's agreement.
The statement says that under the dispute settlement system - set out in the original main deal - the independent arbitration panel (comprising senior judges) could rule that the European Union is acting in such a way as to make the backstop last indefinitely.
Mr Cox told MPs: "There is no ultimate unilateral right out of this arrangement".
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"The attorney general has confirmed that there have been no significant changes to the withdrawal agreement despite the legal documents that were agreed last night", Labour Brexit spokesman Keir Starmer said.
"One door has closed but other possibilities have opened up and markets are hopeful that Wednesday's vote on a no-deal Brexit will suffer a big defeat", said Timothy Graf, head of macro strategy at State Street Global Advisors in London.
Mr Cox's advice was released as Mrs May's agreement was being scrutinised by the so-called "Star Chamber" of Brexiteer lawyers convened by the European Research Group (ERG) of Leave-backing Tory MPs.
The immediate reaction was cautious from the Northern Irish Democratic Unionist Party (DUP) which props up May's minority government.
EU Commission president Jean-Claude Juncker made a bad-taste smutty innuendo about Theresa May as he addressed MEPs today, hours after had agreed changes to her Brexit deal.
The EU said it had made significant concessions as two additional documents were agreed to back up the withdrawal agreement struck in December past year - a joint legally binding instrument which the United Kingdom could use to start a "formal dispute" against the EU if it tried to keep the United Kingdom tied into the backstop indefinitely and a joint statement committing both sides to find an alternative to the backstop by the end of the Brexit transition period of December 2020.
Theresa May's hopes of pushing her renewed Brexit deal through Parliament have been dealt a crushing blow after the Attorney General refused to change his legal opinion on the United Kingdom being able to leave the Irish backstop.
Lawmakers rejected the deal 391-242, ignoring May's entreaties to back the agreement and end the political chaos and economic uncertainty that Brexit has unleashed.
Lawmakers, who rejected the proposal by 149 votes, will get to vote on Wednesday on whether to rule out a no-deal Brexit.
And if that is turned down MPs will have a third vote in three days on whether to extend Article 50 and delay Brexit.
Environment Secretary Michael Gove said it was "make your mind up time" for lawmakers.