• It's been eight years since just under 52% of the British electorate voted to leave the EU. It was a monumental decision that divided friends, families and the nation. It still does, to a lesser extent.
• Brexit has barely been mentioned in the U.K. election race, with neither broadly pro-Remain Labour nor the pro-Leave Conservatives wanting to remind voters of their respective positions on the thorny subject.
• Labour says it wants to "reset" ties with the EU, but has ruled out rejoining the single market or customs union.

It's been eight years since just under 52% of the British electorate voted to leave the European Union, in a monumental decision that divided friends, families and the nation.

It still does, to a lesser extent, with polls suggesting that the same vote held now might produce a different result and constituents questioning whether life outside the economic and political union is really better.

Still, "we are where we are" is the attitude of many Britons, with parties of all political colors vowing in the years between the referendum and the actual January 2020 exit, to "make the best of it."

But if Labour — a party which opposed Brexit but pledged to "respect the will of the British people" — wins the forthcoming U.K. election on July 4 as polls suggest, some analysts predicting a thawing of icy relations between the EU and U.K. and, perhaps, even a rapprochement.

For its part, Labour is treading carefully around the thorny subject of Brexit and any post-election win relationship with the EU, eager not to scare the horses — that is, prospective swing voters — ahead of July 4.

Instead, Labour says it will not attempt to rejoin the single market or customs union that characterize the EU's economic framework that facilitates trade between member states — but signals that it wants to improve relations with Brussels.

Instead, the party has said it wants to reduce trade restrictions and a new veterinary agreement to relax checks on agri-food goods (such as health certificates for products of animal origin, which require a veterinarian's signature). It also says it wants to reduce barriers for touring artists and musicians, and to secure a mutual recognition agreement for professional qualifications "to help open up markets for UK service exporters."

Labour's reticence over making much of its potential post-election plan vis-à-vis the EU, the U.K.'s largest trading partner as a bloc, is largely seen to be down to its eagerness to target disaffected Conservative voters.

Voter polls have consistently pointed to a significant win for the party, suggesting it could clinch around 40% of the vote, compared with 20% for the incumbent Conservative Party led by pro-Brexit Prime Minister Rishi Sunak.

Around 16% of the vote is expected to go to the Reform UK party helmed by Nigel Farage, a politician on the fringes of British politics who has never served as a member of Parliament, but who was, arguably, the rocket fuel behind the Brexit vote.

Hard ball

While Labour will look to improve relations with the EU if it wins the election, analysts say the bloc is hardly likely to fall over itself to make life more pleasant for the U.K. The EU has so far always stuck to its position that Britain would not be able to "cherry-pick" the advantageous bits of its former EU membership that it would like to keep.

After all, the EU has problems of its own, with a rise in hard-right governments and fractious intra- and inter-EU relations as a result. In addition, it's tackling questions over its global economic position, undocumented immigration and war on its doorstep in Ukraine.

"We will have a U.K. government that will wish to change the relationship [with the EU], whether it will succeed is another matter," top U.K. polling expert John Curtice told CNBC.

"What they're wanting to do is to soften the relationship while still being outside any of the former institutions, and the question is to what extent the union will regard it is interested to accommodate that ... partly because they've got other things to worry about, and partly because they may just think: 'Well, you made your bed, you lie in it.'"

Nonetheless, Curtice held there was a shared interest in finding a joint strategy with regard to the high number of undocumented migrants continuing to come into Europe, and across the English Channel to the U.K.

"They might think it's in their interest to reach an agreement about what we do about migrants coming across the EU. Probably, as has been intimated in the past, only by us being willing to accept that we're going to have to be part of the European wide scheme that deals with the resettlement of migrants who come across the Mediterranean. There's going to be a quid pro quo," he said.

Matt Beech, director of the Centre for British Politics at the University of Hull, said he is convinced that Labour Party Keir Starmer will seek more than a rapprochement with the EU.

"I think it would be the heartbeat of Keir Starmer and his close team to get Britain back into Europe, and to ride roughshod over the democratic will of the British people," he told CNBC last week.

"I think that the likes of Starmer and others see membership of the EU as making Britain somehow more virtuous. I don't think that's correct, and I don't think the majority of people in this country thinks that's correct."

CNBC contacted the Labour Party for a response to the comments, and it pointed to its manifesto for its stated position on the EU.