Sorry for long post. This is actually what the highway code says to cyclists on shared paths,
Cycle Tracks. .... Cycle tracks may run alongside footpaths or pavements and be separated by a feature such as a change of material, a verge, a kerb or a white line. You MUST keep to the side intended for cyclists as the pedestrian side remains a pavement or footpath.
Some cycle tracks shared with pedestrians will not be separated by such a feature. On such shared use routes, you should always take care when passing pedestrians, especially children, older or disabled people, and allow them plenty of room. Always be prepared to slow down and stop if necessary
AND
Sharing space with pedestrians, horse riders and horse drawn vehicles.When riding in places where sharing with pedestrians, horse riders or horse drawn vehicles is permitted, take care when passing pedestrians and horse riders, especially children, older adults or disabled people. Slow down when necessary and let them know you are there; for example, by ringing your bell (it is recommended that a bell is fitted to your bike), or by calling out politely.
Remember that pedestrians may be deaf, blind or partially sighted and that this may not be obvious.
Do not pass pedestrians, horse riders or horse drawn vehicles closely or at high speed, particularly from behind.
AND to PEDESTRIANS on shared paths:
Routes shared with cyclists. Cycle tracks may run alongside footpaths or pavements and be separated from them by a feature such as a change of material, a verge, a kerb or a white line. Such routes may also incorporate short lengths of tactile paving to help visually impaired people stay on the correct side. On the pedestrian side this may comprise a series of flat-topped bars running across the direction of travel (ladder pattern). On the cyclist side the same bars are orientated in the direction of travel (tramline pattern).
Some routes shared with cyclists will not be separated by such a feature allowing cyclists and pedestrians to share the same space. Cyclists should respect your safety (see Rule 62) but you should also take care not to obstruct or endanger them. Always remain aware of your environment and avoid unnecessary distractions.
I can't see anywhere where it says cyclists should give way to pedestrians in this situation (as opposed to at side roads). It says they should be careful when passing (but presumes they will pass), leave room and slow down, and let them know you are there with a bell. And that pedestrians should not obstruct cyclists and should be aware of the environment.
EDIT - AARGH I'm wrong, it's in rule H2. Cyclists should give way to pedestrians on shared use cycle tracks and to horse riders on bridleways. But all the other points still stand.