More than two times as many people trust the Conservatives to deliver the best Brexit deal for the UK as Labour, according to a new Opinium/Observer poll. The survey comes before a keynote speech on Brexit to be delivered by Theresa May at Lancaster House in London on Tuesday. It shows the Conservatives also enjoy stronger backing than Labour over the key issue of immigration.

The findings will sound alarm bells within Labour before two impending byelections, the first in Copeland, Cumbria, and the second in Stoke-on-Trent Central, triggered by Labour’s Tristram Hunt stepping down. Hunt announced on Friday that he is to become director of the Victoria and Albert Museum.

The contest in Stoke-on-Trent, a seat Labour has held since 1950, is likely to see strong challenges from Ukip and the Tories. Almost 70% of people in the constituency backed Brexit in last June’s referendum and the issues of Europe and immigration are bound to be centre stage.

Opinium found that 30% of people said they most trusted the Conservatives to deliver a successful Brexit, compared with just 13% for Labour and 11% who cited Ukip. A further 36% of respondents said they did not trust any of the main parties or did not know. On immigration, 22% trust the Conservatives most, with Ukip second on 17% and Labour third on 14%. Among Leave voters, 34% trust Ukip most on immigration, 25% trust the Tories most and only 7% Labour.

Wes Streeting, Labour’s Ilford North MP, says ‘now is not the time for people to quit’ the party. Photograph: Linda Nylind/The Guardian

Labour MPs expressed surprise and shock at the resignation of Hunt, who was a prominent figure on the modernising right of the party. In an article for the Observer, Wes Streeting, the Labour MP for Ilford North, says now is not the time for people to quit the party as it faces populist threats and huge issues such as Brexit. “This isn’t the time to walk away from politics,” he says. “Those of us who want to see a genuine alternative to the ugly populism of Donald Trump and Nigel Farage have to roll our sleeves up and get stuck into the battle of ideas.”

Former shadow cabinet member Chris Leslie said it would be appalling if Labour lost Hunt’s seat: “Stoke has been Labour since 1950. It is a bedrock of Labour support. To lose it would be a catastrophe and would have to provoke a serious rethink about where Labour is going.” Birmingham MP Jess Phillips said she expected Labour to win the seat because Ukip was not well organised in the area, and while many people voted for Brexit, they would not be ready to vote Ukip. “If we lose it will be shocking and I will have been proved wrong,’ she added.

The Opinium poll suggests that if a general election was called tomorrow, the Tories would gain 38% of the vote, giving them a commanding lead over Labour, which is on 30%. Ukip and the Liberal Democrats were both up one point, putting them on 14% and 7% respectively.

The poll also found that if the government had to choose between staying in the single market or ending free movement, 41% would opt for ending free movement, while 32% would choose continued single market membership.

Just over half of UK adults agreed that having some restrictions on the overall numbers of migrants from the EU would not damage UK businesses. However, 40% think businesses will suffer if they cannot hire any migrant labour from the EU – although 32% disagree.

Overall, however, Brexit and immigration have slipped in perceived importance compared to health and the NHS, as the crisis in the healthcare system deepens. The proportion of respondents viewing the NHS as the most crucial issue facing the country rose eight points to 61%, while immigration fell two points to 41% and Brexit one point to 36%.

However, despite 47% agreeing that the Red Cross was right to say the NHS was in a “humanitarian crisis”, voters still believe the health service is better off under Theresa May and the Conservatives than it would be under Jeremy Corbyn and Labour, a separate opinion poll has found.

A ComRes poll for the Sunday Mirror and the Independent found 43% thought the Tories would make a better job of managing the NHS this winter compared to 31% who thought Labour would. The findings will be seen as a further blow for Corbyn as the NHS crisis is being used by the party as a central theme in its campaign to retain Copeland at the upcoming by-election following Jamie Reed’s resignation.