Trzaskowski, the candidate of the opposition Civic Platform party, was backed by 48.37 percent of voters, public broadcaster Polish Radio’s IAR news agency reported.
Celebrity journalist Szymon Hołownia won the support of almost 17.5 percent.
Incumbent conservative President Andrzej Duda, who is standing for a second term in office, was backed by 15.54 percent of voters in Britain, IAR reported.
Amid a coronavirus pandemic, voting by Poles in Britain was by mail-in ballot only.
Duda will face challenger Rafał Trzaskowski in a runoff election next month after no candidate won an outright majority in the first round of voting on Sunday, according to final results announced by electoral officials in Warsaw.
With 100 percent of the ballots counted, Duda, an ally of the ruling conservative Law and Justice (PiS) party, garnered 43.5 percent of the vote overall, electoral officials said on Tuesday morning.
Trzaskowski, the mayor of Warsaw, was runner-up with 30.46 percent overall, according to the National Electoral Commission (PKW).
In the United States, most Polish expats who voted in the first round backed Duda. In Ireland, meanwhile, Trzaskowski finished first, with Duda in third place.