The Hypnotism Act 1952 does not cover broadcasting, but there are other laws which do.
The language in the 1952 Act is about "an entertainment to which the public are admitted", and letting local authorities regulate such events in the same way that they regulate similar events. If you are recording or streaming an event that takes place in Great Britain, then the Act might cover the event itself but not what happens with the resulting video. Some readers may recall Paul McKenna doing TV hypnotism in the 1990s, without having to apply to every council in the UK for permission.
Ofcom's Broadcasting Code includes a provision
2.9: When broadcasting material featuring demonstrations of hypnotic techniques, broadcasters must exercise a proper degree of responsibility in order to prevent hypnosis and/or adverse reactions in viewers and listeners. The hypnotist must not broadcast his/her full verbal routine or be shown performing straight to camera.
This applies to TV and radio broadcasters regulated by Ofcom, per the Communications Act 2003. Video sharing platforms are covered by another code, and video-on-demand services by another. These do not mention hypnotism explicitly, but generally refer to classifications which the British Board of Film Classification would apply to a video recording. For hypnotism, they say:
The BBFC is occasionally required to classify video works which claim to show self hypnosis techniques to help the viewer give up smoking; lose weight; become more confident etc. The BBFC treats these works with caution and has sought expert advice to help with the age rating decision. The Hypnotism Act deals specifically with hypnotism as part of a stage act and so, arguably, does not apply to so called 'self-help' works. The BBFC tends, with some exceptions, to rate these works 18 for an adult audience.
The situation is not entirely clear-cut, but it seems plausible that some material featuring hypnotism should be gated for over-18s, depending on the exact nature of the content. It may tip over the line to "material that might impair the physical, mental or moral development of under-18s", or not. These provisions only matter for services which are regulated by Ofcom.
For Internet-based video more generally, there is a current draft Online Safety Bill which may eventually end up changing the regulatory landscape.