Jeremy Corbyn has avoided the drubbing some had predicted as Labour retained control of key councils in England.
But the party had a bad night in the Scottish Parliament elections, where Nicola Sturgeon won an historic third term for the SNP.
Labour could be pushed into third place behind the Tories in Scotland, with the Lib Dems also staging a revival.
UKIP gained council seats, came second in two by-elections and looks set for their first Welsh Assembly seats.
The London Mayoral result will not be declared until late Friday. Counting gets under way later in 40 English local council polls, with further results to come.
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Labour MP John Mann - one of Jeremy Corbyn's biggest internal critics - conceded that the party was "doing OK" but he said it should be winning by a landslide against a feuding and weakened Conservative Party.
BBC analysis suggests the Tory vote is down in southern England, which may have helped Labour to retain control of councils in areas such as Crawley, Southampton, Norwich and Hastings, where its vote was looking vulnerable.
Labour's vote share is down about 6% on average on 2012 - the last time the seats in England were contested - but a BBC analysis suggests its share is up 4% on the general election in key wards, with the Conservatives down by a similar amount.
It is a different story in Scotland, where the party's share is down 11% and could come fourth behind the Lib Dems, according to one projection.
The party gained one seat from the SNP - Edinburgh Southern but Scottish Labour leader Kezia Dugdale was left relying on the top-up system for a seat, after failing to take Edinburgh Eastern.
SNP leader Nicola Sturgeon is expected to lead her party to a third successive term in office, telling supporters: "We have made history".
But BBC analysis suggests that with more than half of seats declared, its share of the vote is simply holding steady with what it achieved in 2011.
In other developments:
- Shadow Home Secretary Andy Burnham considers running for newly created role of Greater Manchester mayor
- UKIP comes second in two by-elections and gains English council seats including six seats in Thurrock UKIP
- Gill Furniss, the widow of Labour MP Harry Harpham, retains his Sheffield seat in a by-election
- The Lib Dems regain ground in Scotland, with Scottish leader Willie Rennie taking a seat from SNP
- Liverpool Mayor Joe Anderson is re-elected
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The Conservatives have won the Holyrood seat of Eastwood from Labour and Scottish Conservative leader Ruth Davidson won Edinburgh Central from the SNP, taking the Tories to five constituency seats, their strongest showing for some time.
Conservative Commons leader Chris Grayling told the BBC she had been successful because there was "no other political leader in Scotland who is so closely associated with the unionist cause".
Ms Davidson said: "There are people right across Scotland who are sending the SNP a message - the voices and the decision we made as a country will not be ignored."
The Lib Dems held Orkney and Shetland and took Fife North East from the SNP. The SNP took Rutherglen from Labour.
In a surprise result in Wales, Plaid Cymru leader Leanne Wood unseated Labour's Leighton Andrews in the Rhondda - described by former Labour cabinet minister Peter Hain as "a terrible shock for us in Welsh Labour".
Ms Wood said: "A new dawn has already broken over the Valleys, people have voted for change."
Labour's vote is down by eight points overall, the Conservative vote is down by three points, while Plaid Cymru is up by two points. UKIP has seen its vote increase by 12 points, suggesting it will maintain the level it achieved in 2015 which should be enough to enable the party to win its first assembly seats.
Labour has held the Westminster seat of Sheffield Brightside in a by-election, on a 5.8% swing from the Conservatives, with UKIP coming second. The party also held Ogmore in another Parliamentary by-election, in South Wales, with UKIP again in second place.
Image copyright PAResponding to claims Labour is not doing as well as it should be, Shadow Chancellor John McDonnell told the BBC: "We are laying the foundations for a four-year programme."
"If you compare, from when Jeremy took over to now, we are on a clear path of improvement. That's exactly what our strategy is all about."
He said in Scotland the picture was "complex" and Labour was "at the early stages" of rebuilding support having been "wiped out" in 2015.
In total, 2,747 seats in English councils - spanning metropolitan boroughs, unitary authorities and district councils - were up for grabs. The majority of the seats were last contested in 2012.
Thursday's polls are the single largest test of political opinion before the next general election, which is scheduled for 2020, with 43 million people entitled to take part.
What to expect when (all timings are estimates)
Friday 6 May
Around 80 English local election results expected overnight
Scottish Parliament - results in all 129 seats expected overnight
Welsh Assembly - results in all 60 seats expected overnight
Liverpool mayoral result also expected overnight.
11:30 First results from around 50 councils in England to declare on Friday
15:00 First results of London Assembly elections
17:00 First preference vote share information for elections to Northern Ireland Assembly
1800 Result of first preference votes for Mayor of London (final result expected early evening)
Results of Police and Crime Commissioner elections in England and Salford mayoral contest are also expected on Friday.
Saturday 7 May
14:00 Results from five remaining councils in England
16:00 Result of Bristol Mayoral contest
Result of elections to Northern Ireland Assembly expected
Sunday 8 May
16:00 Result of elections to Bristol Council (final council in England to declare)
Results of Police and Crime Commissioner elections in Wales