The Tory party moved the writs for by-elections in the west London seat and in the North Yorkshire constituency of Selby and Ainsty.
Nigel Adams, a close ally of Mr Johnson, has quit as the local MP in this area.
The by-elections are now set to take place between 21 and 27 working days from the writs being moved.
They are therefore due to happen before the Commons rises on July 20.
But ex-Culture Secretary Nadine Dorries, another one of Mr Johnson’s close allies, had as of Wednesday morning still not officially resigned as MP for Mid-Bedfordshire despite saying she would do so.
Downing Street described this situation as “unusual” and stressed that Rishi Sunak believed that the people of Mid-Beds “deserve proper representation” in the Commons.
The Prime Minister’s press secretary said: “We are moving on with the two by-elections of Uxbridge and Selby, I can’t pre-empt what the MP for Mid Bedfordshire is going to do.”
Asked if there is frustration over the delay, she said: “It’s obviously unusual to have an MP say they will resign with immediate effect and for that not to take place.
“The Prime Minister believes the people of Mid Bedfordshire deserve proper representation in this House and he looks forward to campaigning for the Conservative candidate in the by-election.”
Asked if Ms Dorries’ constituents are currently being properly served, the press secretary said: “The Prime Minister believes that it’s important that they have certainty.”
Ms Dorries not resigning may cause more political woes for the PM and stop him being able to call all three by-elections on the same day.
Tory peer and pollster Lord Robert Hayward told The Standard: “All three of these by-elections are in play.
“All parties will treat them as very serious contests.”
He added: ““In a difficult set of circumstances, he (Mr Sunak) has been trying to show that he is in charge of the Government and will therefore want the best possible results in each of the constituencies.
“There is no question that these are important for Keir as well because he has to be able to show that the Labour Party is making progress across the country.”
Holding onto Uxbridge and South Ruislip, where the Tory majority in 2019 was 7,210, could prove more challenging than in Selby and Ainsty, where the majority was 20,137.
But the Conservatives have a history of by-election successes in the west London seat, with Mr Johnson’s predecessor John Randall, now a peer, winning there in 1997 shortly after Tony Blair’s landslide victory.
His predecessor Sir Michael Shersby was also elected at a by-election in 1972.
Demographic changes in the constituency, though, may make it harder for the Tories to win this time.
The Tory majority in Mid-Beds was 24,664 in 2019.
Responding to the writs being moved, Liberal Democrat Deputy Leader Daisy Cooper MP said: “From Uxbridge to Selby people are sick to their back teeth with Conservative MPs taking areas for granted and not listening to local communities.
“Rishi Sunak and his chaotic Conservative Government have let people down so badly, whether it’s soaring mortgages and food prices or the crisis in our NHS.”
Lord Hayward believes that in the short-run Mr Johnson’s resignation means the Prime Minister is facing a “difficult combination of events”.
But he thinks that the ex-PM’s departure may in the long-run turn into Mr Sunak’s “advantage”, partly as it will weaken his support in Parliament.