Thomas Stewart Baker (born 20 January 1934) is an English actor and writer. He played the fourth incarnation of the Doctor in the science fiction television series Doctor Who from 1974 to 1981.

https://en.wikipedia.org/wiki/Tom_Baker

Doctor Who is a British science fiction television series broadcast by the BBC since 1963.

https://en.wikipedia.org/wiki/Doctor_Who

What's the joke?

The play on words is obvious.

David John Tennant (né McDonald; born 18 April 1971) is a Scottish actor. He is best known for portraying the tenth incarnation of the Doctor in the sci-fi series Doctor Who

https://en.wikipedia.org/wiki/David_Tennant

Over the last 200 years, the pronoun whom has been on a steady decline. Despite its waning use in speech and ongoing speculation about its imminent extinction, whom still holds a spot in the English language, particularly in formal writing.

https://www.thesaurus.com/e/grammar/whom-vs-who/

You use who in questions when you ask about the name or identity of a person or group of people.

Example sentences: * Who's there? * Who is the least popular man around here? * Who do you work for?

https://www.collinsdictionary.com/dictionary/english/who

You use whom in questions when you ask about the name or identity of a person or group of people. Example sentences: * 'I want to send a telegram.'—'Fine, to whom?' * Whom did he expect to answer his phone? * 'You're too sensitive.'—'Too sensitive for whom?'

https://www.collinsdictionary.com/dictionary/english/whom

Charles, Prince of Wales is the former title of Charles III (born 1948), before his accession to the throne of the United Kingdom and the other Commonwealth realms.

https://en.wikipedia.org/wiki/Charles,Prince_of_Wales(disambiguation)

Charles III (Charles Philip Arthur George; born 14 November 1948) is King of the United Kingdom and the 14 other Commonwealth realms.

https://en.wikipedia.org/wiki/Charles_III

Joseph Ellis Pasquale (born 20 August 1961) is an English comedian, actor and television presenter.

[...]

He won the fourth series of I'm a Celebrity...Get Me Out of Here! in 2004 and participated in the eighth series of Dancing on Ice in 2013. He also hosted the revived version of the game show The Price is Right.

https://en.wikipedia.org/wiki/Joe_Pasquale

Queen are a British rock band formed in London in 1970 by Freddie Mercury (lead vocals, piano), Brian May (guitar, vocals), and Roger Taylor (drums, vocals), later joined by John Deacon (bass).

https://en.wikipedia.org/wiki/Queen_(band)

Sir Brian Harold May CBE (born 19 July 1947) is an English musician, songwriter, record producer, animal rights activist and astrophysicist. He achieved worldwide fame as the lead guitarist and backing vocalist of the rock band Queen

https://en.wikipedia.org/wiki/Brian_May

The museum has over 300 restored exhibits of garden machinery from over the last 200 years, as well as a collection of lawnmowers previously owned by famous people including Prince Charles and Princess Diana, guitarist Brian May, performer and presenter Paul O'Grady, comedian Lee Mack, and Coronation Street actress Jean Alexander, who lived in Southport for many years.

https://en.wikipedia.org/wiki/British_Lawnmower_Museum

dibber definition

a tool used for making holes in soil when planting seeds or small plants

https://dictionary.cambridge.org/dictionary/english/dibber

Trowel definition

a small tool consisting of a flat metal blade joined to a handle, used for spreading building materials such as cement

https://dictionary.cambridge.org/dictionary/english/trowel

The word "tool" could mean different things:

Tool definitions:

a piece of equipment that you use with your hands to make or repair something

OR

(offensive) an insulting word for a person who you dislike very much or who behaves very stupidly

OR

(offensive) a penis. https://dictionary.cambridge.org/dictionary/english/tool

So "Famous tools" could mean:

Equipment (tools) that are recognised by many people.

OR

Famous people who are disliked very much or who behave very stupidly.

OR

Famous penises a.k.a. dicks, dongs, knobs...

Where's the joke?

I think the audience laughed because it sounds like Lee could be talking about looking for famous penises on a website (or I just have a dirty mind).

Frederick "Fred" Flintstone is the main character of the animated sitcom The Flintstones, which aired during prime-time on ABC during the original series' run from 1960 to 1966

[...]

Fred's trademark catchphrase yell is "yabba dabba doo!"

https://en.wikipedia.org/wiki/Fred_Flintstone

Listen to how Fred says Yabba Dabba Doo! on YouTube

Where's the joke?

This catchphrase "yabba dabba doo" sounds similar to the question "What does your dibber do?".

Lee is asking Dale if he is Fred Flintstone because he sounded like him when he asked the question "What does your dibber do?"

Inspector Morse is a British detective drama television series based on a series of novels by Colin Dexter.

https://en.wikipedia.org/wiki/Inspector_Morse_(TV_series)

The British Lawnmower Museum is a museum dedicated to the history of the lawnmowers in Southport, Merseyside, northern England.

https://en.wikipedia.org/wiki/British_Lawnmower_Museum

The nation's foremost garden machinery collection, multi-award winning unique museum of over 250 restored machines.

https://www.tripadvisor.co.uk/Attraction_Review-g191255-d564490-Reviews-British_Lawnmower_Museum-Southport_Merseyside_England.html

Would I Lie to You?...

(abbreviated as WILTY) is a British comedy panel show aired on BBC One.

For each show, two celebrity guests join each of the team captains. The teams compete as each player reveals unusual facts and embarrassing personal tales for the evaluation of the opposing team. Some of these are true, some are not, and it is the panellists' task to decide which is which.

[...]

In all rounds, the scoring system is the same: teams gain a point for correctly guessing whether a statement is true or not, but if they guess incorrectly the opposing team gets a point.

https://en.wikipedia.org/wiki/Would_I_Lie_to_You%3F_(British_game_show)

'possessions' element was introduced, in which the panellist takes an item out of a box and reads a statement from a card, and has to convince the opposing team that the possession genuinely belongs to them. https://en.wikipedia.org/wiki/Would_I_Lie_to_You%3F_(British_game_show)