The pre-1960 license (correct UK spelling, but I don't really know why) exemption to which you refer is to do with the vehicle itself and its requirement to have some form of Vehicle Excise License (road tax) and other vehicle related documents. Sadly it has nothing to do with driving licences (correct spelling when refering to driving licence, again I don't know why).
Depending on when you were issued with any type of driving lecence will dictate what size of motorcycle you can or cannot ride with it.
Check this web page and follow the various links as required
http://www.direct.gov.uk/en/Motoring/DriverLicensing/WhatCanYouDriveAndYourObligations/DG_4022547There is also the problem with people that did have a motorcycle entitlement on their licence suddenly losing it when they applied for a new photo licence.
It is far too complicated to work out what licence you need without being in possesion of all your details including inside leg measurement but that link above should put you on the road to righteousness.
To cut a long story short, you need a motorcycle entitlement on whatever driving licence and you possess and then some......