The Scottish Parliament on Tuesday voted in favour of another referendum, although 59 voted against it, meaning the first minister won the battle for a second chance at independence.
Despite Ms Sturgeon's minority government winning the vote, the UK Government reiterated now is not the time to hold a divisive vote.
A UK Government spokeswoman said: "The Prime Minister has been clear that now is not the time for an independence referendum, and we will not be entering into negotiations on the Scottish Government's proposal.
"At this point, all our focus should be on our negotiations with the European Union, making sure we get the right deal for the whole of the UK.
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Lord McConnell warned against a second referendum becoming a distraction
"It would be unfair to the people of Scotland to ask them to make a crucial decision without the necessary information about our future relationship with Europe, or what an independent Scotland would look like.
"We have been joined together as one country for more than 300 years. We've worked together, we've prospered together, we've fought wars together, and we have a bright future. At this crucial time we should be working together, not pulling apart.”
Just hours before the vote was held, a former first minister slammed Nicola Sturgeon for dragging Scotland back into an another independence battle.
Speaking to the European Union Select Committee, Lord McConnell ripped into the SNP for using Brexit to further the bid for a second referendum.
I think any move towards another referendum needs to be treated cautiously
The Labour peer said: "I mean the first thing I would say is the prospect of another divisive referendum in Scotland is not something that cheers my heart.
“Apart from a few people who enjoyed the debate, you know and they tended to be activists, taking part in creative discussions amongst themselves, I don't think the last referendum was a wholly enjoyable experience for people across Scotland.
“I came across many people who found the whole experience really quite difficult, so I think any move towards another referendum needs to be treated cautiously and I think not just for its division but for the very real need to get the timing right.”
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Lord McConnell said he was worried about the question of a second independence referendum being a distraction from preparations for governing Scotland post-Brexit.
“My biggest worry about this is that they, both Governments, ministers in both governments and maybe some civil servants as well, spend most of the rest of this year preparing arguments about this issue instead of preparing the necessary legislation, executive orders and all the other things that will be required in order to make Brexit work for Scotland in a practical sense,” the former first minister said.
“I am seriously concerned about this debate around the referendum, however predictable it might be is going to be huge distraction from the urgent business, because there is a time limit on this now, sorting out the legislation, the capacity, the arrangements that are going to be required for Scotland to be governed properly, regardless of the constitutional position following the final day of Brexit in two years time.”