You can take your bike on this commuter service, which costs either £3 or £5 per crossing, depending on whether you believe the photocopied timetables at the hotel or the signboard at Canary Wharf, and runs all day every day, every few minutes at rush hours, less regularly off-peak.
There are several of these commuter ferry services, operated by Thames Clippers. They can take you and bike across the Thames, between the piers of say Tower Bridge's north bank and London Bridge's south bank.
But they don't really count as 'Thames Crossings' any more than taking your bike on an overground train service would. The Hilton-Canary Wharf service is the only water service that explicitly traverses the Thames between facing piers, and the boats clearly say FERRY on the side, so it's a legitimate crossing for our purposes.
Personally I'd miss it out as it costs five quid that could be spent on a perfectly good packet of crisps or glass of coke in a Docklands bar, but here it is listed for completeness.
On Westferry Circus, the roundabout just behind the Canary Wharf Pier, is the Traffic Light Tree. The artwork, from 1998 by Pierre Vivant, is presumably a metaphor for every financial expert around here. It's financed by the taxpayer, does what it likes, and sends out mixed signals.
From here on, either north or south bank, it's streets roughly paralleling the riverside, or the promenade itself, to Greenwich. A Google map shows both routes between Tower Bridge and Greenwich. Either way it's about two miles to Greenwich Foot Tunnel.