What a confusing world... Some of these names are included in OS Open Roads, so I'll work through removing any other names the Queen won't (yet) let us have!

(I also used old OS maps via NLS and iD)

Hi ndm. Sorry, I'm still getting to grips with OSM! This isn't a direct import. I viewed the South Glos council data by imprting it as a custom layer on iD, and then went over the map individually adding the footpaths to match satellite imagery, street level photos, OS OpenData, and my own knowledge having grown up in the area walking and cycling these paths. In future I'll break things up into smaller edit sets. I'll try and give it all another look over, and flag up any issues.

Sorry, I should have broken this into seperate edits which specific references. Primary source is S Glos's planning consultations on bus lanes, pedestrian crossings etc https://consultations.southglos.gov.uk/consult.ti/system/findConsultations and planning permissions re the Metrobus Extension: https://www.southglos.gov.uk/transport-and-streets/transport/planning-transport-policy/cribbs-patchway-metrobus-extension/
Pegasus Rd
This for example shows Pegasus reaches all the way to Standing Stone roundabout:
https://consultations.southglos.gov.uk/consult.ti/PT.4279PatchIghPX/consultationHome
Hayes, Merlin and Highwood around San Adreas is clarified in the Metrobus planning. Eg: https://www.southglos.gov.uk//documents/San-Andreas-general-arrangement-drawing.pdf
San Andreas, Standing Stone, Christmas Tree, and the Cone roundabouts are all included in planning consultations. Oddly I already knew the names of the latter three, so maybe they used to be signposted? San Andreas was recently in the Evening Post... Presumably a very slow day! https://www.bristolpost.co.uk/news/bristol-news/ever-heard-san-andreas-roundabout-3086824
Lysander Road with B&Q and Morrisons is perhaps questionable though, as this is based primarily on the addresses of all the businesses there listing Lysander Rd (exc Tile Flair - which predates the road). Otherwise the road has no name.