The "B" stands for baroque, which is the fingering system the instrument uses. The baroque system is the most common fingering system for the recorder, as opposed to the slightly simpler (but generally not as good overall) German system—see the video in this answer: Is it difficult to transition from German fingering to Baroque fingering?
To play the C major scale (the easiest scale) on a soprano/descant recorder with baroque fingering is relatively straight forward:
There is only one note that has a somewhat odd fingering in this scale and which requires a "forked" fingering. (The German system makes this particular note easier to finger, but to do this compromises have been made in the design that mess up intonation and various chromatic notes.)
For the larger alto/treble recorder, this same fingering pattern produces an F major scale:
To be able to play both alto and soprano recorders requires learning the fingering for each size of instrument separately. This can be confusing at first, but (with practice) it's not as hard as you might think.
do re mi fa so la tiis solfege. In some cultures do always refers to C. If that is not what you mean then you should be able to use the same fingering charts for all recorders.