I have some time, but never very much…
beaucoup
adverbe
a. [modifiant un verbe] a lot, a great deal
b. [dans des phrases interrogatives ou négatives] much, a lot, a great deal
is there any left? — not much = est-ce qu’il en reste ? — pas beaucoup
© Larousse 2014
The French translation sounds off here.
would feel a bit less weird, but still… Not great.
I don’t know. In French, It kind of feels like we’re saying the same thing twice.
Also, “jamais”/never “bothers me”, because I can’t really think of a situation in which we would actually say that.
have a slightly different meaning (compared to the English sentence), but they feel more natural somehow. (Probably because I could see them actually being used in some contexts).
“Beaucoup” can be used in affirmative (positive) sentences as well, for instance:
I think the translation is not completely incorrect, but does sound a bit off. What I would change instead is the order of pas and toujours. So:
J’ai un peu de temps, mais pas toujours beaucoup…
Initially I thought the French translation may have been AI/machine generated, but after looking at the English sentence, I decided that it didn’t really make sense.
“Oh…” - That was my reaction when I read your post 
- because I didn’t even consider the possibility.
It does sound much more natural indeed.
Maybe that’s what the original translator had in mind and they inadvertently swapped both words…
@yottapolyglot
Added some formatting to clarify. I had put both because the meaning changes a little in interrogative or negative phrases. Thank you for emphasizing the difference.