The rules for Reported Speech apply here.
The original utterance or thought is
What is going to happen when we meet the next time.
When you report this thought, backshifting the verbs is normal, and this applies not only to meet but also to is going. So
I was worried, thinking what was going to happen when we met the next time.
Even though when you meet the next time is still to come, this is fine. This is sometimes called future-in-the-past (Cambridge dictionary), since you are talking about a future event using past tense forms.
However, when you use reported speech and the event is still true from the speaker's viewpoint, you can retain the present tense. Thus you can say
I was worried, thinking what is going to happen when we meet the next time.
Since the next time you met has not yet occurred, the speaker can use the present tense (is going and meet).