From what I have found it all depends on what the specific telehandler is registered as. For example it could be registered as agricultural plant or a tractor, which from memory has age-related weight limits, however it can all be done on a standard B car licence. There are other categories that it could be registered as a mobile crane which then would mean you would need to have other categories on your licence (C).
However, it's worth bearing in mind that depending on what the telehandler is registered as there could be certain conditions that need to be complied with. Agricultural registration would limit the road activities to those solely for agricultural or forestry purposes and there would also be distance limits that you can't exceed.
It is a complete minefield, I'm not an expert at all, just many hours of reading and trying to better understand the regulations. Best to speak to DVSA (I believe) if you need to clarify certain technicalities - that being said you would need to speak to somebody there that properly understands it.
