http://www.travelinescotland.com/cms/Travelbybike.xhtml has a really comprehensive list of transport providers – bus, rail, ferry – that offer bike carrying facilities.
http://www.scotrail.co.uk/content/cycle-rescue.html operates an excellent cycle rescue scheme where cyclists who hold a valid rail ticket, but cannot reach their destination because of accident, vandalism or irreparable breakdown, will be transported to the nearest ScotRail-served station.
http://www.ramblers.org.uk/info/britain/public-transport-for-walkers.htm is a really useful leaflet with details of public transport services for walkers throughout the country, as is
http://www.keswick.org/visitor-information/transport-in-keswick/ for the North Lakes
http://www.cornwallpublictransport.info gives information for Cornwall
Some other authorities and transport users’ groups producing details of rural bus services, some of which are specifically designed for walkers, include:
http://www.northyorkmoors.org.uk/moorsbus/ in North Yorkshire
http://www.pcnpa.org.uk/website/default.asp?SID=1166&SkinID=3 in Pembrokeshire
http://www.dalesbus.org/ (includes details of the Dales Bike Bus) in the Yorkshire Dales
http://www.staffordshire.gov.uk/transport/publictransport/connect/ (can carry up to 4 bikes) in Staffordshire Moorlands
There are more around the country. Check first with the relevant local authority and tourist information office.
Use http://www.traveline.org.uk and http://www.transportdirect.info for route options and times.