As noted on way/18835045 , cycle access as mapped in OSM "just stops" here on the motorway. Imagery suggests that a separate path exists across the motorway bridge. I didn't notice signage when last here (but wasn't looking for it); but the cycle route past the Millennium Hotel is signed.
Probably needs local knowledge or a survey to say what is best here.

There's a gap in the Teesdale Way at relation/957011#map=17/54.544006/-1.929340 / Does anyone know what route it currently takes there?

According to http://osm.mapki.com/history/node.php?id=27365306 , Westminster has spent most of its time as a town rather than a city (especially when it has last been added by locals).

Currently it's a "city" - is that correct?

(the confusing thing here is that "city" in OSM terms means "calls itself a city in real life" - see place=city for the discussion about that)