Originally Posted by ISTFlyer

I'm sorry to tell you that airlines are not obliged to pay for private transportation when there is a publicly available method ( in your case train ). Some airlines do for goodwill purposes.
Here, in your situation, Qatar Airways have offered to pay for your train costs to Cardiff and you simply rejected for personal reasons ( fear of COVID ) which is acceptable. However, this does not mean that you could take a taxi and ask QR to pay for it.

Plus, you have accepted to be transported to London rather than Cardiff, QR did not force you to travel to Heathrow; you could have simply rejected traveling to Heathrow and ask for an alternative travel route. Possibly a supervisor or a local QR office could have rerouted you BKK-DOH-AMS on their own aircraft and AMS-CWF on KLM if you have requested it. Regular phone agents do not have the power to rebook you on other airlines.
By the way, it's obvious that they wouldn't rebook you LHR-AMS-CWF separately after you agreed on travelling to Heathrow rather than Cardiff.

I am not writing this post to say that you have done something wrong and it's your fault; I am writing this because you would get nothing from the small claims court if you sue them as you personally agreed to be transported to London yourself.