As you shift octaves, all the notes cross hands and may be played with different fingers
That does sound a bit weird. But:
As you shift keys, all the fingerings shift (relative to the tonic)
That sounded a bit scary to begin with, but on reflection it's exactly what I'd expect. I learned to play (flute) from dots years ago; I can solfege a bit, but I still basically think chromatic rather than diatonic - I mean, I think of G-B-D-g and D-F#-A-d as two different series of notes, rather than as the same thing in two different keys.
On English concertina you have to finger each note in a chord separately for all chords
Again, that's what I'd expect - I had to read this twice to work out what the problem might be.
Piers:
if you don't want to put the effort into learning to play it well (though that doesn't mean you have to learn to play complicated things), it would be better not to do it at all ... playing an instrument at the same time will probably make your singing worse.
I'm not really bothered about getting good enough to play tunes on the concertina - I've got whistles for that. The goal will be to learn to play it well enough that I can concentrate on singing. I think that'll be enough of a challenge to keep me going.