Note that the standard fingering for low C (one ledger line below the staff) is 0. Now move two measures to the right and note that the fingering for D is 1-3. If you see a C in your music, but play with fingering 1-3, you are playing your Bb instrument in C.
Consequently, you can create a fingering chart for your Bb instrument to play in C by moving all the fingerings on the chart by two measures to the left. C# becomes 2-3, D becomes 1-2, etc.
Using these fingerings you can play with piano or organ without transposing (you really are transposing, but you don't have to think about it). But the down side is you can't play a duet with another trumpet, unless you transpose down a step or he transposes up a step.