My name is Fran, and I am a US lawyer.
Yes, if the company will not resolve your problem for you, you can try to recover from your credit card carrier. However, if you purchased this time share in 2006, don't expect to get satisfacton. Generally, they will only reimburse for a finite amount of time, usually one or two billing periods.
If you get no satisfaction from your credit card carrier, the next step would be to contact a Florida lawyer to write a strong letter to the company, threatening all appropriate legal action if they will not resolve this matter and give you what you contracted for your money back. Even if you don't go on to sue, sometimes that can be enough to bring about some kind of a settlement.
From there, start involving the US consumer protection agencies we have, as they are free. Go to the Better Business Bureau's site and lodge a consumer complaint against them. You can do that right on the BBB's website.
As a non-profit organization committed to protecting consumers, the BBB will try to intervene on your behalf and to resolve your problem if they can locate the company and the company is willing to work with them.
From there, you can also involve the various government agencies that protect consumers by filing formal consumer complaint reports. You can do this on line and would want to file one with the Florida State ATtorney General's Office
which is where someone would go to lodge an official complaint against a Florida company. You want to emphasize that the company has breached its contract with you and engaged in fraudulent, unfair and deceptive marketing practices.
Finally, you can involve our Federal consumer protection regulatory agency, the Federal Trade Commission
by filing a similar consumer complaint with them also. These government agencies will not usually intervene on any one person's individual situation. But when they get enough complaints to see a pattern and practice of unethical or fraudulent business dealings they can, and do step in. They haul the company into Federal court where the companies usually settle for enormous fines and heavy government sanctions. The money the government collects is distributed among those defrauded by the company. So even if it takes a while for you to recover, you want to be file these complaints.
You also can hire a Florida consumer right's lawyer and sue for fraud or breach of contract, and one might be willing to take the case on contingency, meaning that you would only pay the lawyer if he wins your case or gets a settlement you approve of. Then he gets about 40% of your award. You could sue from the UK but then even if you won the judgment you'd have to get in enforced in a US court, which is not always as easy as it sounds.
These are all of your civil options. If the company is an out and out scam and you can't even find them, which doesn't seem to be the case, or you want to threaten criminal action, then the matter becomes a job for law enforcement. You would need to report them to your police and make out a report. You would also have to report them to the police in Florida in the jurisdiction where they are, because it is the place where the crime was committed that controls in a criminal case.
As this is an international matter, you can report this as well to the FBI. You can use their Internet Crimes Complaint Center online at IC3.gov.
I will be the first to recognize that none of the above provides a perfect solution, as I am sure you want your money back sooner rather than later, and this is going to be a slow head-banging-against-the -wall process before you can get anything back even under the best of circumstances.