More English election results are set to be announced on Saturday, giving the parties a clearer idea of how many councils they will end up controlling.
The Conservatives have already made significant gains, with Labour losing control of several local authorities.
Counting is also under way in English mayoral elections.
PM Boris Johnson called the results so far "encouraging", while Labour leader Sir Keir Starmer promised to lead a fightback and set out a "bold vision".
On Thursday, people in England voted in elections for 143 councils, 13 mayors and 35 police and crime commissioners.
This represents the biggest test of public opinion since the 2019 general election - expanded further by some of the contests being postponed from last year because of the pandemic.
The results are set to continue coming in over the weekend, with Covid restrictions causing delays and counting centres working according to different timetables.
On Friday, the Conservatives celebrated winning the Hartlepool parliamentary by-election, taking a Westminster seat which Labour had held since its creation in 1974.
Mr Johnson's party also picked up control of several councils, including Maidstone, Cornwall, Nottinghamshire and Basildon, while Labour lost some, including Sheffield and Plymouth, to no overall control.
And the Conservatives managed to keep the Tees Valley mayoralty - in an area once considered a Labour stronghold - with Ben Houchen increasing his share of the vote to 73%.
The party gained new police and crime commissioners, for Cleveland, Avon and Somerset and Dorset.
It is expected the London mayoral result will be announced on Saturday, although if it is close it could take longer.
Results so far
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Mr Johnson called the Conservatives' results thus far "very encouraging" but cautioned against too much celebration before all of the outcomes were known.
Sir Keir offered a stark assessment of Labour's performance, which follows its worst general election outcome since the 1930s in 2019.
He said his party had "lost the trust of working people" and pledged to take "responsibility for fixing things".
Note: This lookup covers national elections in Scotland and Wales, the Hartlepool by-election, as well as council and mayoral elections in England and Police and Crime Commissioner (PCC) elections in England and Wales. There may be parish council elections or council by-elections where you are. Check your local council website for full details. Last updated: May 7, 2021, 14:22 GMT
But Labour won some key votes, including in Doncaster, where Ros Jones was re-elected as mayor, and retained control of councils including Liverpool, Manchester, Hastings and Exeter.
Liberal Democrat leader Sir Ed Davey, whose party has lost a few council seats, said it was "moving forwards and making progress".
The Green Party has picked up seats, the party's one MP, Caroline Lucas, saying it felt like a "green spring is happening".