A groupset must be more than £135? in which case for me this is one of the situations that need more digging before ordering because you could be heading for the double-VAT situation and a job to claim it back.
I just had a look, initially I could get to checkout with VAT still added on, as you say. But then I seemed to set a country somewhere and suddenly items were not available for
UK GB shipping - I saw the attached screenshot, despite the cheerful banner at the top saying UK shipping was open:
If they are shipping items >£135 with VAT still attached I could guess a few things...
1 - They are set up for the <£135 scenario but still charging VAT for >£135 - my understanding is this leads to double-VAT and a job to claim it back
2 - They have a UK subsidiary and can import via that (a bit like UK companies are scrambling to set up EU subsidiaries for the reverse direction) and so can indeed charge VAT for >£135 and the transaction will be hassle free. They are no longer Wiggle are they, if they were maybe that would do it. I think the new owner - Riverside - is German but could still have UK subsidiaries set up already?
3 - My understanding is still wrong and there is something else going on...
Also, that point 2 about importing via a subsidiary is similar to the rule about marketplaces like ebay and amazon - they are allowed to do the VAT for all transactions regardless of its value. The definition of a marketplace was reasonable, something that does more than just sell their own stuff, but also collects money on behalf of other retailers. So buying through amazon and ebay should be straightforward.
UK / Great Britain / England is not listed on this page, which defines their VAT (I would have thought footnote 2 applies to UK) - https://www.bike24.com/shipping.html ... can't find anything else on their site apart from they sound confident about shipping to the UK