I notice that Barnet libraries have decided that a 30 minute public transport journey is a reasonable distance to go to reach your nearest library. Lincolnshire, in their libraries consultation which was overturned by judicial review, argued that a 30 minute car journey was a reasonable test. Both of these seem less generous than the standard adopted in Brent 's Libraries Transformation Project which saw 97% of the population within 1.5 miles of a Brent library. Even just walking that would take an able bodied person not more than 30 minutes. By either public transport, cycling or car it would take rather less.
Of course distance and journey time are not the only relevant tests of accessibility, but it seems to me that Brent's six library network is probably one of the most accessible in the UK.