The Conservatives had faced a particularly tough task because the last time the seats were contested was in 2015, on the same day as elections for the national Parliament in which the party — then led by David Cameron — won a surprise victory.

Labour, in contrast, did not do so well in 2015, and this time around, experts said, its path should have been easier. But early evidence suggested that it had not succeeded in appeasing either supporters of Brexit in the north and in the middle of the country, or opponents in big cities and in the southeast.

Labour had said that it would support a second Brexit referendum, but only in some circumstances, and that it wanted to negotiate an alternative exit deal.

Barry Gardiner, who speaks for the party on international development, told the BBC that there were “two competing principles here, and we are trying to hold them in tension.”

Though Labour believes that there is a way to reconcile them, Mr. Gardiner admitted, “if a party is seen to be speaking with two voices, it’s very difficult to communicate the policy.”

The parties will now gear up for elections to the European Parliament on May 23, when two new parties that did not run on Thursday will take part: the Brexit Party, which backs leaving the European Union quickly and without an agreement if necessary; and Change UK, which wants to remain in the bloc.

The Brexit Party, led by Nigel Farage, the former leader of the United Kingdom Independence Party, is being particularly closely watched — recent opinion polls suggest it is tied with Labour for the lead in those elections, or even ahead, and taking much of its support from former Conservative voters.

In Thursday’s local elections, the United Kingdom Independence Party, which remains a hard-line pro-Brexit party but has taken a turn to the right under its new leader, Gerard Batten, lost more than 50 seats.