I used this list. Adds up to 40. Whether the list is correct is another matter.
Manager Start Date - End Date Record
Tommy Wright 19 Apr 2021 - Present Won: 2, Drawn: 0, Lost: 2
John Sheridan 13 Nov 2020 - 18 Apr 2021 Won: 8, Drawn: 4, Lost: 21
Noel Hunt 04 Nov 2020 - 13 Nov 2020 Won: 1, Drawn: 0, Lost: 2
Richie Wellens 13 Nov 2018 - 04 Nov 2020 Won: 35, Drawn: 17, Lost: 31
Phil Brown 12 Mar 2018 - 11 Nov 2018 Won: 10, Drawn: 11, Lost: 11
David Flitcroft 05 Jun 2017 - 01 Mar 2018 Won: 21, Drawn: 3, Lost: 18
Luke Williams 30 Dec 2015 - 05 May 2017 Won: 20, Drawn: 22, Lost: 33
Martin Ling 03 Nov 2015 - 29 Dec 2015 Won: 5, Drawn: 0, Lost: 4
Lee Power 18 Oct 2015 - 03 Nov 2015 Won: 0, Drawn: 1, Lost: 2
Mark Cooper 24 Jul 2013 - 17 Oct 2015 Won: 52, Drawn: 28, Lost: 45
Kevin MacDonald 01 Mar 2013 - 12 Jul 2013 Won: 4, Drawn: 5, Lost: 5
Fabrizio Piccareta 19 Feb 2013 - 01 Mar 2013 Won: 1, Drawn: 1, Lost: 1
Paolo Di Canio 20 May 2011 - 18 Feb 2013 Won: 54, Drawn: 18, Lost: 23
Paul Hart 03 Mar 2011 - 28 Apr 2011 Won: 1, Drawn: 4, Lost: 6
Danny Wilson 26 Dec 2008 - 02 Mar 2011 Won: 44, Drawn: 40, Lost: 37
David Byrne 14 Nov 2008 - 26 Dec 2008 Won: 1, Drawn: 4, Lost: 3
Maurice Malpas 15 Jan 2008 - 14 Nov 2008 Won: 13, Drawn: 11, Lost: 18
David Byrne 27 Nov 2007 - 15 Jan 2008 Won: 3, Drawn: 4, Lost: 3
Paul Sturrock 07 Nov 2006 - 27 Nov 2007 Won: 26, Drawn: 11, Lost: 15
Adrian Williams 24 Oct 2006 - 07 Nov 2006 Won: 0, Drawn: 0, Lost: 2
Dennis Wise 22 May 2006 - 24 Oct 2006 Won: 9, Drawn: 5, Lost: 3
Iffy Onuora 26 Sep 2005 - 22 May 2006 Won: 10, Drawn: 15, Lost: 15
Andy King 20 Dec 2001 - 26 Sep 2005 Won: 71, Drawn: 48, Lost: 75
Roy Evans 03 Aug 2001 - 20 Dec 2001 Won: 10, Drawn: 6, Lost: 10
Andy King 01 Nov 2000 - 31 Jul 2001 Won: 12, Drawn: 11, Lost: 14
Colin Todd 04 May 2000 - 01 Nov 2000 Won: 5, Drawn: 6, Lost: 9
Jimmy Quinn 02 Oct 1998 - 02 May 2000 Won: 19, Drawn: 21, Lost: 45
Steve McMahon 28 Nov 1994 - 23 Sep 1998 Won: 74, Drawn: 48, Lost: 79
John Gorman 04 Jun 1993 - 21 Nov 1994 Won: 14, Drawn: 20, Lost: 34
Glenn Hoddle 04 Apr 1991 - 04 Jun 1993 Won: 49, Drawn: 32, Lost: 34
Ossie Ardiles 19 Jul 1989 - 31 Mar 1991 Won: 38, Drawn: 33, Lost: 31
Lou Macari 23 Jul 1984 - 03 Jul 1989 Won: 132, Drawn: 63, Lost: 75
Ken Beamish 01 Mar 1983 - 30 Jun 1984 Won: 24, Drawn: 17, Lost: 28
John Trollope 01 Nov 1980 - 01 Mar 1983 Won: 45, Drawn: 36, Lost: 44
Bobby Smith 17 May 1978 - 31 Oct 1980 Won: 62, Drawn: 26, Lost: 46
Danny Williams 06 Mar 1974 - 01 May 1978 Won: 87, Drawn: 61, Lost: 78
Les Allen 10 Nov 1972 - 28 Feb 1974 Won: 13, Drawn: 20, Lost: 31
Dave Mackay 31 May 1971 - 01 Nov 1972 Won: 18, Drawn: 18, Lost: 25
Fred Ford 01 Oct 1969 - 31 May 1971 Won: 35, Drawn: 23, Lost: 25
Danny Williams 01 Aug 1965 - 01 Jul 1969 Won: 104, Drawn: 58, Lost: 60
Wiki has it at 47 (again no idea if its true), the turn over since 2000 is mind blowing!
Danny Williams 19651969
Fred Ford 19691971
Dave Mackay 19711972 (player-manager)
Les Allen 19721974
Danny Williams 19741978
Bobby Smith 19781980
John Trollope 19801983
Ken Beamish 19831984
Lou Macari 19841989 (player-manager)
Osvaldo Ardiles 19891991 (player-manager)
Tony Galvin 1991 (caretaker)
Glenn Hoddle 19911993 (player-manager)
John Gorman 19931994
Andy Rowland 1994 (caretaker)
Steve McMahon 19941998 (player-manager)
Mike Walsh 1998 (caretaker)
Jimmy Quinn 19982000 (player-manager)
Colin Todd 2000
Andy King 20002001
Roy Evans 2001
Andy King 20012005
Iffy Onuora 20052006
Dennis Wise 2006
David Tuttle 2006 (caretaker)
Ady Williams 2006 (caretaker)
Paul Sturrock 20062007
David Byrne 20072008 (caretaker)
Maurice Malpas 2008
David Byrne 2008 (caretaker)
Danny Wilson 20082011
Paul Hart 2011
Paul Bodin 2011 (caretaker)
Paolo Di Canio 20112013
Fabrizio Piccareta 2013 (caretaker)
Tommy Miller & Darren Ward 2013 (joint player-caretakers)
Kevin MacDonald 2013
Mark Cooper 20132015
Lee Power 2015 (caretaker)
Martin Ling 2015
Luke Williams 20152017
David Flitcroft 20172018
Matt Taylor 2018 (player-caretaker)
Phil Brown 2018
Richie Wellens 20182020
Noel Hunt 2020 (caretaker)
John Sheridan 20202021
Tommy Wright 2021 (caretaker)