The video is using prboom, on that engine you can edit the hud via the -PRHUD- lump in prboom-plus.wad .
Prboom+, Crispy Doom, Chocolate Doom & CnDoom are the most used and recommended engines for speedrunning on the main Classic Doom leaderboards if not using vanilla.
On Zandronum, some features can be done with a custom SBARINFO lump, such as an InGame Timer:
https://zdoom.org/wiki/DrawString
https://zdoom.org/wiki/SBARINFO
example of usage: " DrawString SMALLFONT, white, time, 200 + center, 0, 1; "
This can be used on any server by having your custom hud's wad in your skins folder.
There is however a speedrun wad for Zandro that was being developed in 2016:
https://zandronum.com/forum/viewtopic.php?t=7418
That one of course will not work via skins folder and needs to be hosted server-side. Supports database to save times.