In mainland Britain, you are never more than 61 miles from a bike rack.
This fact is calculated from data recorded in OpenStreetMap.
The point that achieves the furthest distance from a bike rack is here, with the nearest to it being this.
If you know of a bike rack that disproves the claim, you can edit OpenStreetMap (here's how) to make sure that item is listed on the map. (NB this website is refreshed occasionally from OpenStreetMap, but not instantly.)