This is a problem of translation because there are distinct Greek words for 'bride' and 'bridegroom'. John the Baptist showed this (when denying that he was 'the bridegroom' - the Messiah) in John 3:29. If we turn to that first, then all should become clear. John the Baptist said:
"He that hath the bride is the bridegroom..." where 'bride' is
numphe, but 'bridegroom' is numphios (Young's Concordance, p116)
There is yet again a different Greek word for bride-chamber, which is in Mark 2:19-20 in translations that stick to the Received Text. The two translations you use are not literal translations. Indeed, 'The Message' is not really translation, but a rendering of the Bible in vernacular English language of today, intending to give the meaning as people today might best understand it. It follows the 'dynamic-equivalence' method of treating scripture. The NLT combines that with a bit of formal-equivalence. Neither stick to the Received Text. It is the Authorised Version, and Young's Literal Translation that stick purely to the Received Text, giving formal-equivalence (literal translation). That is why the A.V. and Y.L.T. bring in the word 'bridechamber' at Mark 2:19, for this third Greek word is in the Received Text:
"And Jesus said unto them, Can the children of the bridechamber fast,
while the bridegroom is with them? " (A.V.)
"And Jesus said to them, 'Are the sons of the bridechamber able, while
the bridegroom is with them, to fast?' (Y.L.T.)
This third Greek word is numphon. Yet in no Greek manuscripts can the word for bride - numphe - be found in the two verses you ask about. This immediately removes the problem you raise. There is no problem when the exact Greek words are exactly translated. As you have pointed out, only 'The Message' introduces 'the bride' into the text, but it should not.
Having cleared that up, the next point to make is the context, where Jesus was addressing a question raised as to why the disciples of John the Baptist, and the Pharisees, fasted, but his disciples did not fast. That is in Mark 2:18. Jesus then replied that whilst he (the bridegroom) was in their midst, those around him would celebrate - not fast! When the time came that the bridegroom would be taken away from them, then his disciples would fast.
This leaves the rest of New Testament scriptures about the future marriage of the Lamb to his 'bride' to be understood as Christians have always understood them. The Church that Christ builds, which the gates of hell cannot overcome, will be united to him in heaven at that glorious marriage supper, after the old heaven and earth have passed away, and God has created a new heaven and a new earth in which righteousness will dwell. Then the Bride appears, as the city of New Jerusalem, with 12 gates having the names of the 12 tribes of Israel, built upon the 12 foundations of the Apostles of Christ. This shows that all who had saving faith in both Old and New Testament eras, form that heavenly city, which is also described as a woman, the wife of the Lamb.
To answer your main question: No, the wedding had not yet started in Mark 2:19-20. But the bridegroom had arrived on earth, and all who recognised him as such rejoiced in his presence. They celebrated and did not fast while he was with them. When he was taken away from them, then the situation changed, but they had the promise of being reunited with him in heaven. Then, once all had been resurrected, judged and assigned their eternal portion, the symbolic 'bride' of Christ would experience that heavenly 'consummation' of their union with Christ.