On Monday evening (November 4) Sir Lindsay was elected as Speaker of the House of Commons, replacing former Speaker John Bercow.
He beat his fellow Labour MP Chris Bryant by 325 votes to 213 in the final of four rounds of voting.
Following his election, the Liberal Democrats candidate for Chorley, Paul Valentine, announced he would not oppose Sir Lindsay's re-election in the December 12 general election, in line with parliamentary convention allowing the Speaker to stand unopposed by major political parties.
Speaking about the party's plans prior to the election, Mr Valentine said: "We knew a general election was in the air but we also knew that once John Bercow said he was going to resign from politics that Lindsay was going to throw his hat in the ring to replace him.
"We have had to prepare for both eventualities [Lindsay winning and losing]. But we have wanted to progress the Liberal Democrats locally anyway for a long time while also being respectful if we weren't to be running.
"As a main party we believe in upholding traditions. We stand by the speaker being elected unopposed.
"There's no exceptional circumstances here to change that.
"He has to resign from his party and be independent. We are very satisfied that he will be an independent member and be neutral in the House of Commons.
"We also expect him to keep representing Chorley."
Sir Lindsay previously told the Lancashire Post that he will keep serving Chorley if he was elected as Speaker.
"From our position there's no criticism in terms of Lindsay as a person. In terms of our outlook if we were running against him, we were of the belief to 'play the ball, not the man'," Mr Valentine added.
"We would have criticised the leadership in the Labour Party and local issues but it would have been no reflection on Lindsay.
"I don't know him personally but from everything I have heard from council members and constituents they all praise his work ethic and how much his office does for Chorley.
"Going after him was never going to be our plan of attack. We would have brought a positive Liberal Democrat agenda which is very different offer from other parties at the moment."
Mr Valentine said the local party's eyes are now on the 2020 local elections, where all councillors in Chorley will be re-elected due to boundary changes.
He said: "Our long term plans look at the local elections next year with all the councillors up.
"We think from our point of view we will keep the momentum going.
"We want to give residents an option to choose from and hopefully we can give them a positive alternative.
"When you have a single party dominating the chamber [Labour currently has 37 of 47 seats in Chorley] scrutiny suffers."
Following Sir Lindsay's election, Green Party candidate for Chorley James Melling said he has "every intention of representing Chorley Green Party in this election".
Chair of the local party, David Harrison, said: "We are in the throes of a Climate Emergency and the 'business as usual' attitude of the major parties cannot be allowed to continue.
"Labour and Conservative governments have consistently made the crises worse with policies based on economic growth.
"Our planet is in jeopardy and with it the lives of our children and grandchildren."
Mark Smith, the Brexit Party candidate for Chorley, has announced he will not be contesting the seat against the new Speaker.
The Conservative and Labour parties have not yet publicly announce if they will contest the seat.