They don't have to be registered Robert, but do technically need single vehicle approval. However, this was only ever done on one pedelec to my knowledge at the only inspection site that would do it, at Southampton. Before and since then many ride around with a twist throttle and nobody looks or cares, least of all the police. So long as your assist speed is limited to 15.5 mph and the motor is 250 watt rated or less they won't bother you.
Getting legal permission to have a fully acting "Twist and Go" throttle on a pedelec:
The Motorcycle Single Vehicle Approval test now includes a sub-category for "Twist and go" electric bikes, which require type approval as Single Vehicle Approval (SVA), but which can otherwise be used like other normal pedelecs. Fewer modifications will now be needed for these machines to pass the test and gain type approval. The MSVA test costs £55 per individual vehicle.
Each individual bike must pass the Motorcycle Single Vehicle Approval (MSVA) test. This provides UK-specific type approval (i.e. it is not valid in other EU countries) and the DfT regard these as legal bureaucracy free pedelecs.
The new class for Twist & Go EAPCs is called "250W LPM", a sub-category of the standard "Low Powered Moped", which is the L1e-A category which T&G EAPCs fell into before. The new edition of the test manual reduces the requirements somewhat for this new sub-category. For example, a kickstand is still required, but it does not have to auto-retract. Moped-like mirrors are now not required, etc., etc. Basically almost any legal pedelec meets the requirements, but check the PDF manual linked to below:
https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/821685/motorcycle-single-vehicle-approval-inspection-manual.pdf
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