It was quite hard to find (didn't seem properly geolocated on the map), but here are all the planning documents and drawings for the new station; https://citydev-portal.edinburgh.gov.uk/idoxpa-web/applicationDetails.do?previousCaseUprn=000906466691&previousCaseType=Property&previousKeyVal=KMS443EW03300&activeTab=summary&previousCaseNumber=KMS43MEW03300&keyVal=KR8VH7EW7N000 (or search 09/02589/FUL)
There's a drawing for the underpass, which is 5.5m wide but appears to have tactiles at either end and glass sliding doors at the station end. Crucially, the access statement from Network Rail says "Cyclists will be required to dismount, but cycles can be taken through the underpass.
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Interestingly, the drawing also states that "only one pedestrian crossing is provided at the new interchange to deter pedestrians from walking in a westerly direction towards the tram depot and away from the interchange". You wonder if that's a deliberate attempt to design the underpass as station access only, and not as a safe and convenient route around the roundabout and under the A8 for through cyclists/pedestrians.
Further looking through the drawings, there will be a path connection from the A8 pavement (north side) to station, but that ends in "dismount" signs at the station. Interestingly, there appears to be segregated ped/cycle path running right up to the entrance to the underpass on the station side (but with dismount signs for the underpass itself). So you might have to push through the underpass, but it looks like it does provide an off-road link between the Gyle, under the A8 to the pavement route along the north side of the A8.
There looks to be an aspirational path along the northern boundary of the tram depot from the "West Edinburgh Development Area", but the pavement around the northern perimeter of the roundabout is marked only as "pedestrian route to RBS".