-게 되다 vs -게 되었다
되다 has many meanings, but in your example with future action, it means "be arranged/decided" (결정되다, 정해지다).
The -었다 ending indicates a resulting state (like present perfect) of the action rather than one in the past.
되다 is the base form (no conjugation) of the verb, which is seldom used is in normal conversation or writing. The present tense form of it is 된다.
So 내일부터 같은 반에서 공부하게 되었어요 means the decision/arrangement has been made for a future course of action.
The other tenses can also be used with -게 되다, with slightly difference meanings.
1. ... 됐어요 (past / present perfect) = It's been decided for me to study in the same class starting from tomorrow.
2. ... 돼요 (present) = I am going to be studying in the same class ... (a little less clear).
3. ... 될 거예요 (future) = I will likely end up studying in the same class ...
Between #1 and #3, -게 되다 has a slightly different meaning - to be arranged/decided for something vs to come to a certain situation. #2 sits in between #1 and #3. It seems not as clear in meaning as the others for this example.