There is one caveat: R2 (and WD) lose their mating potential on 12x12. Adding Camel moves cures this, even up to 16x16 (the largest the Applet can do). So I have little doubt R2-then-B would have mating potential there.
For B2-then-R very large boards actually make it easier. Because from a large distance you could use the R legs as if the piece were a Rook. Even an invulnerable one, because the bare King cannot approach it. And the F step can be used to tighten the noose around the bare King. So it should be a very easy checkmate.
There is one caveat: R2 (and WD) lose their mating potential on 12x12. Adding Camel moves cures this, even up to 16x16 (the largest the Applet can do). So I have little doubt R2-then-B would have mating potential there.
For B2-then-R very large boards actually make it easier. Because from a large distance you could use the R legs as if the piece were a Rook. Even an invulnerable one, because the bare King cannot approach it. And the F step can be used to tighten the noose around the bare King. So it should be a very easy checkmate.