Even with the improvements suggested in other answers:
I hadn't received the grades for one of my courses until last week,
implies you expected your grades before last week, that the receipt of your grades last week was later than expected;
But it doesn't seem to be updated in the system yet,
also seems to be implying an unexpected delay - so why the "but"? This sentence agrees with the preceding one, but "but" is supposed to present a contrast.
You could usefully simplify the first sentence and replace but with and to correctly indicate the two agree; presumably you say your grades were received late to reinforce your assertion about their visibility in the system.
As for the main part of the question, I can either read
it doesn't seem to be updated in the system yet,
it doesn't seem to have been updated in the system yet
(treating updated as the past tense of the verb), or as
it doesn't seem to be up-to-date in the system yet
(treating the adjective up-to-date as an approximate synonym)
If I assume the former, both versions of your second sentence are roughly the same apart from the strength of your assertion (is not compared to does not seem).
If I assume the latter, the first version is entirely in the present tense, whereas the second version is a present observation about something that (did not) happen in the past.
That ambiguity, and the awkwardness mentioned at the top of my post, are more significant than the differences between the two versions. Instead of either, I would prefer a construction like
I only received the grades for one of my courses last week, and they are still not visible in the system.
which moves attention to the consequence of interest, and its presumed cause.