The UK legislation was harmonised with EU Law EN15194 in April 2015 which means that it could change as Brexit takes effect. For now it is pretty clear in defining what can, and what cannot, be called an ebike.
Your steed is an "electrically assisted pedal cycle" (or EAPC, or ebike, or pedelec) if: the bike has pedals that propel it; the electric motor will not assist you if you are travelling at more than 25 km/h (about 15.5 mph) and the power does not exceed 250 watts.
The cycles that meet these requirements can be ridden on any cycle paths and anywhere else that bikes are normally allowed.
In the UK you must be over 14 years old to ride an electric bike, but you do not need a licence, nor do you need to register it or pay vehicle tax.
You may find that some off-road bikes that can go faster than 15.5 mph by flicking a switch, but for UK law these are not compliant with EAPC regulations for on-road use.
If your ebike does not meet the regulations it will need to be registered, insured and taxed as a motor vehicle. In ths case, you will also need a driving licence, and you must wear a motorcycle helmet.
Harmonisation with EU law has had an important effect on electric bikes with twist and go throttles that can propel the bike to full speed without any pedalling at all. From January 1st 2016 the only throttles legal within the UK's EAPC legislation are those that assist the rider without pedalling up to a maximum speed of 6 km/h (about 3.7 mph or walking speed).
If you bought an ebike with a full speed throttle before January 1st 2016 there is no need to panic as these are still considered to be EAPC compliant.