Well ?
🇧🇪 Now and always Unbent, Unbowed, Unbroken.
I think your question is about the cast listing here on IMDb, not the onscreen credits. You're right, Booth is unaccountably missing, as are (at least) the actors who play Narracott the boatman and the announcer who makes the phonograph record thinking it's for a play. Ironically, the announcer asks whether he'll be credited.
These can only be mistakes. IMDb cast listings for series apparently include all speaking parts, annotated to show how many episodes or which seasons a character appears in.
These omissions need to be corrected. At any rate, there's no rationale for omitting Booth.
Oh, and Anna Maxwell Martin (Mrs. Rogers).
There really does seem to be a problem with Episode 1 people, but there shouldn't be. The page is for the whole production.
All of the actors mentioned in this thread are credited on the page for the episode they appeared in, but the main page for the entire series only shows castmembers that appeared in 2 episodes or more.
This is just the way the IMDb software calculates the numbers to create the info on the main page. It's been brought up before and can be especially jarring on mini-series such as this one. But the cast lists on the episodes themselves all appear to be correct and complete.
Chip