The UK's two leading political parties resume talks this week in a bid to find a way out of Britain's Brexit deadlock following a battering at the polls.

Both the ruling Conservative and opposition Labour parties suffered at the hands of voters in "Town Hall" elections for their terrible handling of Brexit. They failed to agree on a way ahead for any aspect of Britain's referendum vote to leave the European Union.

The Conservatives lost control of 45 councils and 1,269 council seats. Labour, despite losing less than the Conservatives, also saw a net loss of six councils and 63 seats, according to the final result reported by the Guardian.

The elections were focused on local issues, and ostensibly had nothing to do with Brexit. However, considering the all-pervading presence of Brexit, it became a harsh barometer of public mood.

UK Prime Minister Theresa May said losing more than 1300 local council seats proved the country wanted to "get on with Brexit."   

Town Hall councilors, the foot soldiers of British politics, reported hostility on the doorstep from both those campaigning to leave the EU and those who want Britain to remain.

Tony Travers, a professor of politics at the London School of Economics told CCTV: "It's unusual for both of these parties to do badly in the same set of elections. And what that tells us is that people are fed up with both the ruling Conservative Party and the opposition Labour Party, partly I suspect about Brexit."

The Conservatives and Labour now appear chastened by the collapse in support and have promised to return to talks Tuesday in an urgent bid to resolve their differences and get back to politics as usual.

But that promise has brought further apparent problems for the two sides, as their supporters, both in Parliament and in the country, have warned against party leaders stitching up a deal together.

If May and opposition leader Jeremy Corbyn were to ignore the warnings, they'd risk splitting their parties down the middle and causing possibly irreparable damage.

The Leavers maintain a forced marriage of electoral convenience would leave a post-Brexit Britain still subject to EU rules and without the ability to negotiate its own trade agreements around the world.

The Remainers claim a Brexit fudge between May and Corbyn would deprive them of a so-called People's Vote endorsing any deal that's agreed, which is a Labour Party policy.

Both the big beasts of British politics say they've heard loud and clear the message from the voters.

Their problem is they are not even close to finding an answer to that call and a route out of the Brexit impasse.

The only politicians in Britain with a smile on their faces at the moment are the Independent Candidates in last week's local elections and the Liberal Democrats.

The Liberal Democrats normally struggle at every level of British politics, but this time they secured their best result in 15 years.

Independent candidates now have representatives in Town Halls across the land in unprecedented numbers.  

It would appear, the ability for Brexit to upend UK politics shows no sign of waning and, consequently, politicians of every hue are turning a wary eye to the European Parliamentary Elections toward the end of May.

Coming three years after Britain voted to quit the EU, these are the elections that should never really happen.

The received wisdom is that if the Town Hall elections were tough, just wait until voters vent their fury next time round.