If I buy a bike from a store and take the bike permanently out of the UK, can I recover the VAT?

No. Under the VAT Retail Export Scheme, customers leaving the EU for more than 12 months were able to reclaim VAT on the bike by presenting a form VAT407 to customs upon exporting the goods from the EU. Unfortunately, as part of a review of duty-free shopping, obtaining this refund process ceased on 1st January 2021.