Electric and non-electric cycles are treated equally. Folded cycles can be taken on the entire TfL service at any time. For non-folded cycles, some restrictions apply. In general, the rules are:

For tube & overground - bikes are allowed Monday to Friday before 07:30, between 09:30 and 16:00, and after 19:00. And anytime on weekends and bank holidays. However, this does vary between lines.

For buses - non-folded cycles are not permitted.

For full details see here.

E-scooters are banned on all TfL services. This not only includes the likes of trains and buses but extends to stations and other premises on the network as well. 

Why are e-scooters banned on the tube?

This rule was introduced as a safety measure to prevent defective batteries from causing fires on the network. It is permitted to carry a non-motorised scooter on the network however it must be able to fold. 

Attempting to take an e-scooter on the tube has a maximum fine of £1,000. More information can be found here.