How do each of these people connect to Doctor Who?
As you might expect, it's a long list.
- Sean Pertwee is the son of the late Jon Pertwee, who played the Third Doctor
- Olivia Colman appeared in "The Eleventh Hour" as the Mother, one of the forms taken by Prisoner Zero
- Peter Davison played the Fifth Doctor
- Louis and Joel Davison are Peter Davison's sons and David Tennant's brothers-in-law
- Jenna Coleman played companion Clara Oswald in series 7, 8 and 9
- Matt Smith was the Eleventh Doctor
- Steven Moffat is a long-time writer for the revived series, and was showrunner from series 5 to series 10
- Heddi-Joy Taylor-Welch was a production runner in series 3 and 4, and was an assistant director from "The Next Doctor" through "The Time of the Doctor"
- Louisa Cavell was assistant director in the second half of series 7
- Lauren Kilcar was a costume assistant on "The Day of the Doctor"
- James DeHaviland has been second assistant director since "Planet of the Dead"
- Janet Fielding played Fifth Doctor companion Tegan Jovanka
- Sylvester McCoy was the Seventh Doctor
- Colin Baker was the Sixth Doctor
- Rhys Thomas has no connection to the show that I can tell; he seems to be a fairly well-known comedian. He's apparently a fan, though, having contributed to the 2012 book Behind the Sofa: Celebrity Memories of Doctor Who
- Georgia Moffett appeared in "The Doctor's Daughter" in series 3, and appeared in the animated miniseries Doctor Who: Dreamland. She's also Peter Davison's daughter and is married to David Tennant
- Olivia Darnley is apparently a friend of Georgia Moffett's
- Nikey Wardley played Tamsin Drew, companion to the Eighth Doctor in the Big Finish audio adventures
- Marion Baker is Colin Baker's wife
- Katy Manning played Third Doctor companion Jo Grant. She reprised the role for an episode of The Sarah Jane Adventures
- Louise Jameson played Leela, companion to the Fourth Doctor
- Carole Ann Ford played Susan Foreman, the Doctor's granddaughter
- Deborah Watling played the Second Doctor companion Victoria Waterfield
- Sophie Aldred played Seventh Doctor companion Ace
- Sarah Sutton played Nyssa, a companion for the end of the Fourth Doctor's era and through much of the Fifth's
- Lalla Ward played the second incarnation of Romana, Time Lady companion to the Fourth Doctor. She was also briefly married to Tom Baker
- John Leeson was the original voice of Fourth Doctor companion K-9, and reprised the role many times since, including the Tenth Doctor episodes "School Reunion" and "Journey's End", spin-off series K9 and The Sarah Jane Adventures, and a handful of promotional appearances on other programs
- Anneke Wills played Polly, companion to the First and Second Doctors
- Lisa Bowerman played Karra, one of the Cheetah people in the final episode of classic Who, "Survival". She also played Seventh Doctor companion Bernice Summerfield on Big Finish
- Matthew Waterhouse played Adric, a companion of the Fifth Doctor
- Paul McGann played the Eighth Doctor
- Jon Culshaw lent his voice to the 2001 BBC audio drama Death Comes to Time. He's also known for an astonishingly good impersonation of Tom Baker1, which would explain the role he plays in the special
- Jemma Churchill had a role in one of the Big Finish dramas
- Lucy, Bindy, Lally, and Rosy Baker are Colin Baker's daughters
- Bruno Du Bois has no connection to Doctor Who that I can tell. He was an assistant director on The Hobbit movies, presumably where he met McCoy (Radagast in those films)
- Peter Jackson is an independent director, famous for black comedies like Braindead and Meet the Feebles; he also made some CGI snooze-fests in the 2000s and 2010s that nobody saw. There have been long-standing rumours that he will one day direct an episode of the series, but as yet has not.
- Ian McKellan is a famous stage actor, and lent his voice to the 2012 Christmas special "The Snowmen"
- John Barrowman played Ninth and Tenth Doctor companion Jack Harkness
- Alice Knight and Sarah Churm have no connection to the show that I can find. Churm has previous appeared alongside Peter Davison
- Nick Jordan and David Kelley were apparently on staff at the Doctor Who Experience in Cardiff
- David Tennant was the Tenth Doctor, and is Peter Davison's son-in-law
- Richard Cookson was script editor for "The Day of the Doctor" (and a few episodes since), and for the docudrama An Adventure in Space and Time, about the initial production of Doctor Who
- Elizabeth Heery is Peter Davison's wife
- Marcus Elliot has no connection I can find
- Ty Tennant is Georgia Moffett's son
- Barnaby Edwards is a long-time Dalek operator, and has lent his voice to several adaptations
- Nicholas Pegg has been a Dalek operator for much of the revived series, and is also very involved in the writing and production of Big Finish audio dramas
- David Troughton is the son of the late Patrick Troughton, who played the Second Doctor. He's also appeared on the show several times, most recently in the Tenth Doctor episode "Midnight" as Professor Hobbes
- Nicholas Briggs is the Voice of the Daleks, and is very involved in Big Finish
- Frank Skinner seems to be a fairly well-known comedian and fan of the show. He has since appeared on the show, but had not done so at the time of The Five(ish) Doctors
- Adam Paul Harvey was formerly in a long-term relationship with Georgia Moffett
- Derek Ritchie was a script editor for "The Time of the Doctor", but had no prior association I can tell
- Michael Houghton has no connection I can tell
- Dan Starkey is most famous for playing Strax the Sontaran Butler/Nurse, but also appeared as Commander Skorr in "The Sontaren Stratgem"
- Russell T. Davies was the showrunner for series 1-4
- Des Hughes was a producer for the second half of series 7
- Gabriella Ricci was assistant production coordinator for series 7
- Sandra Cosfield was has been production secretary since series 7
- Christian Brassington lent his voice to one of the Big Finish audios
Uncredited (but on IMDB)
- Tom Baker was the Fourth Doctor. The archive footage used is taken from the unfinished story "Shada", which was written by Douglas Adams2. The same footage was used to explain Baker's absence in the original "The Five Doctors", back in 1983
- Juliet Cadzow was the voice of the evil Ice Lady in "The Snowmen"
- Ellie and Joseph Darcy-Alden played the children in "The Snowmen"
- John Hurt (who appears in archived sound) played the War Doctor in "The Day of the Doctor"
- Jemma Redgrave has played Kate Stewart, daughter of original series regular Brigadier Lethbridge-Stewart, in several episodes of the revived series
1 Nod to OrangeDog for pointing this out in comments
2 Yes, that one
The Doctor Who wiki gives a comprehensive list of the cameos, and a cross-reference for who they all are.
To summarize, they're all people who have appeared on Doctor Who in some capacity. Sean Pertwee and David Troughton are the sons of Jon and Patrick, and appear in their stead.