Jonboy, from what I can make out from the images below, it would seem the council need to get their signs sorted out. However, these are taken from 2016 Google images so may possibly be out of date and the signs could possibly be different now.
However, from these images the restriction on LRN from Sports Direct southwards indicates 'No Vehicles' (top image) which in my view includes cycles (being ridden). However, as mobility scooters are classed as being useable on footpaths they would be allowed, providing the ones that are allowed on roads are only driven at the 4-mph max that is allowed on footpaths, although I am sure that many of the road ones drive down there in excess of that speed. Should cycles be allowed? in my opinion - No! Mainly because so many ride at speeds well in excess of even the mobility road scooters 8-mph and they weave in and out around the pedestrians at those ridiculous speeds which can only be classed as dangerous.
Having said that, the second and third/fouth pictures below paint a different picture when approaching from Beach Road towards LRN from the Battery Green Road end. The sign in the second picture indicates that cycles are allowed. However, when you get to the junction of LRN (third/fourth images) there is a predicament. Looking north (third picture) there is no sign at all so a cyclist would be in his/her right to ride up there northwards. But looking south (fourth picture) the sign says 'No Vehicles'
Consequently, the second and third pictures contradict the first picture outside Sports Direct for the same stretch of pedestrianised area..
However, as I said at the start these images are from 2016 so they may be different now. I will try and remember to look next time I am there unless someone else does and posts about them here.
edit: just noticed (not visible on the third picture) that there is a 'No Vehicles' sign going north but it is outside the third shop up LRN on the left. Surely that should be at the junction, not so far up the road? Maybe the council couldn't afford to have a post placed there.