A gent with a British accent called and asked for some advice. "My car is sitting in a garage in Naples with a burned-up engine. I drove my Austin Mini down from New England to Miami. In Miami, I had the oil changed at a fast lube facility. I then headed across Alligator Alley toward Naples, and the car began to make noises and the engine finally quit. I had it towed the rest of the way to Naples, where a garage informed me that the engine was ruined because it had no oil in it. I called the fast lube place and was assured that they had put oil in it, as proven by the fact I had made it halfway to Naples. They accept no responsibility for my problem. The garage that has the car has quoted me many thousands of dollars to replace the engine. I don't know what to do. What would you suggest?"
You must have started out with oil, or you would not have gone as far as you did, that much is true. However, your oil went somewhere while you were driving, until there wasn't enough to lubricate the engine, and the engine burned up or seized. The question is where did the oil go, and who, if anyone, is at fault?
The two most likely escape routes taken by your oil are burning or leaking. Burning might be caused by a defect in the engine, like a broken piston ring, and leaking could be caused by someone not tightening the oil drain plug, or the oil filter at that last oil change.
Many people leap to the conclusion that the last mechanic to touch a car has screwed it up and caused new problems. In this case it might actually be true. Ask the Naples garage to look for signs of leakage from the engine. If something was left loose it would have definitely left signs, and the loose part would still be loose. If so, take good pictures, get a statement from the garage manager, and send both to the fast lube company headquarters, not just the branch you dealt with. Hire an attorney to draft a letter demanding a settlement, with a threat to sue if necessary. These companies have insurance for these things, and they will pay if you force it. Simultaneously contact your insurance company and your warranty company and see if they will cover your problem. Maybe they will, and they can extract the money from the fast lube joint.
If the problem was an internal burning of the oil in the engine, the exhaust system will show the contamination. The garage can remove the catalytic converter and inspect it for oil damage. If that is what happened, unless your warranty company will cover it, you, in the American vernacular, are screwed. If you are going to pay for the repair, let me suggest a used engine from a salvage yard, to reduce the cost. I'm sorry for your experience here with your car, and I hope it works out for you.