This is, I think, a three part saxophone arrangement of the theme, with Rollini playing the bass line, making four saxophones. Trumbauer plays the lead part on alto sax with what sounds like a second alto and a tenor behind him. What you hear at 0.56 min. Albert, is part of the lead voice, and also the part which makes it clear that Trumbauer is playing an alto sax and not C melody, as he hits what would be the top note of an alto of that period. I don't hear a clarinet at any point (Murray would be playing tenor sax). At the end everybody is playing. There are two trumpets or cornets but the second one is only audible on the ride out. The only instrument I have problems hearing is the trombone.
A lot of fuss has been made about this recording but I don't think it was at all the disaster a lot of people have made it out to be. The atmosphere in the studio was perhaps tense (one of the vocalists makes a slip with the lyrics) but if Trumbauer had been able to listen to a replay of the recording perhaps he wouldn't have found it so bad after all, or maybe whatever upset him in the first place would have coloured a re-audition also. Musicians, either individually or collectively, are often convinced that a take is bad, but are surprised to hear how good it sounds on listening to it. The reverse is also the case.
Also I think there are many, many, trumpet (or cornet) solos worse than this one. Its in tune, the phrasing is good and in rhythm, and the choice of notes is good.
And the break by Rollini leading into the solo is one of my favourite key change phrases.