The Prime Minister is carrying out talks in the devolved nations ahead of her promise to trigger Article 50 and formally start the process of withdrawing from the European Union by the end of the month.
Brexit minister David Davis has said there would be no sudden drop in numbers, as it would take years to fill low-skilled jobs in hospitality, social care and agriculture now done by immigrants.
Triggering Article 50 will start the clock ticking on a two-year countdown to Brexit and allow negotiations to start between London and Brussels in the coming weeks.
"We are on the threshold of the most important negotiation for this country for a generation", Davis said.
Sturgeon said it was clear that the United Kingdom was heading for a "hard Brexit" to the detriment of Scotland, and that Scottish voters deserved a choice of remaining in the European Union as part of an independent nation.More news: Britain to trigger Brexit process on March 29
Margaritis Schinas, chief spokesman for the European Commission, confirmed officials received May's decision, but would not offer any comments. He noted that negotiations would begin once other European Union states had met to confirm the Commission's negotiating mandate.
"We think the imminent activation of Article 50 will trigger hard trade negotiations, which is not properly priced into the currency", Michael Cahill, New York-based strategist at Goldman Sachs, wrote in a note to clients.
Scottish leader Nicola Sturgeon threw a wrench in those plans last Monday by announcing a push for a new independence referendum from the United Kingdom, which also includes England, Wales and Northern Ireland.
Britain's ambassador to the EU, Tim Barrow, will also play a major role, and the Foreign Office will talk to individual member states to try to get them on its side.
Continued full membership of the customs union is unlikely as it would prevent Britain striking its own trade deals with non-EU countries, a key plank of May's strategy for a new "global Britain".More news: Congress awaits FBI, NSA testimony on Russia-Trump links
Britain believes it can negotiate the exit agreement and a deal on future relations within the two-year negotiating period, although diplomats are sceptical.
The notification of triggering Article 50 will come in the form of a letter.
May has rejected that suggestion, saying "now is not the time" for another referendum on Scottish independence.
But May has repeatedly ruled out an early vote, and on Monday her spokesman told British journalists that there was "not going to be one".
Britain joined in 1973, but has always been a somewhat reluctant member, with a large contingent of euroskeptic politicians and journalists regularly railing against regulations imposed by European Union headquarters in Brussels. For those who would like Britain to stay after all, the 27% unsure of what they would favour in those circumstances does hint that opinion might change if negotiations go badly enough.More news: Doc Rivers hating National Basketball Association back to back scheduling
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