Though it currently doesn’t specify any defaults for the UK. I’d say its worth setting bicycle=no on a footway/path if it is explicitly signed as “no cycling” anyway, just to make it clear.
As for the Garmin maps, that depends on the defaults used by the software to generate them (probably Mkgmap), and any style rules used etc. I think the Mkgmap default rules consider highway=path / highway=footway / highway=cycleway to be much the same in terms of bicycle access. Though some OSM Garmin maps might set this differently.
You could try Openmtbmap which does clever stuff with Garmin bicycle routing, including an option for using footways or not: http://openmtbmap.org/about-2/autorouting/
I won’t see something like that, whether advisory or legal, as bicycle=no. For example you’ll sometimes have a bridge taking a motorway over a river where cyclists must walk their bike on the sidewalk. Yet this may be a major connection in the bicycle network, and it wouldn’t do to have routing software ignore it.
Perhaps you can use bicycle=dismount and if anyone comes up with a better idea they can change it.