Airlines nearly always initially deny EU 261 compensation, hoping you will give up and abandon the claim. Whether you pursue it further depends on how strongly you feel about it, and how much time you are willing to spend on it.
It sound like they at first tried to get you to New Jersey the same day, but the UK passport limits caused that plan to fail.
It will probably take some significant time and effort to fight the EU 261 battle with them, and you may fail. There are some companies which will do the paperwork and fight for you in exchange for a fee. Perhaps that would work for you.
Good luck!