The Financial Ombudsman deals with businesses that provide financial services. I doubt that includes motorcycle dealerships.
Under the Consumer Rights Act, the bike must be of satisfactory quality (taking into account its age and mileage if it was second hand), fit for purpose and match any description that was given. The bike clearly does not meet these requirements. You are therefore able to reject it and receive a full refund. This includes the value of the bike that you part exchanged. You need to get on with this as you only have 30 days from taking delivery to do this.
After 30 days, you must give the dealer a chance to repair or replace the bike - they can choose which. If the repair or replacement fails, you can get the dealer to reduce the price or provide a refund - note that they are allowed to make a deduction for fair use of the bike.
For the first six months, any fault is assumed to have existed when the bike was delivered. If the dealer wants to argue otherwise, they have to prove their case, e.g. by showing that the fault was due to you misusing the bike. After six months, it would be up to you to prove that any fault was present on delivery.
As you paid by credit card, your credit card provider is jointly liable with the dealer, so you can approach them if the dealer continues to ignore you.
I would ring them to see what they say. Follow up with an email to record your understanding of the conversation. I would double check that the email address you are using is correct in case the reason they haven't responded is that they haven't actually received your emails.
If you are getting nowhere with them, send them a letter before action setting out the facts, including copies of any documents you would rely on in court, telling them what you want from them (a full refund, presumably) and giving them a reasonable time to respond. Tell them that, if they fail to respond satisfactorily, you will take legal action without further notice. I would also try your credit card provider in parallel with this.