tl/dr
First 10pm flight from Newark to London Heathrow was cancelled due to a fault, as was the second 6 hours later. Finally got to a hotel at about 9:15. Finally got a flight out from JFK with a different airline nearly 24 hours after the first was due to leave.

Long story:

• 10:30pm United Flight departed gate on time: 10.30pm.
  
• Taxied onto runway and sat still. Approximately 11pm, told technical fault. Towed back to gate to investigate.
  
• Problem not able to be fixed. Told another aircraft would be available shortly. Disembarked approx. 11.30pm.
  
• Gate information suggested departure at 12.30am. Subsequently moved to 1.30am, 2.30am and 3am. * Occasional updates suggesting food transfer to new flight was a problem due to ‘configuration’ of aircraft.
  
• Given $15 of food vouchers on boarding card, however, no food outlets open. Used on chips, chocolate, bottled drinks as this was only option.
  

• Boarded second flight at approx.. 3am. On runway at 3.30pm. Told we were being moved from gate to meet departure slot but paperwork was being completed.

• Shortly after told another technical fault with fuel pump. Towed back to gate to check problem. Could not be resolved. Ground staff informed us there would be a flight put on at 6pm for passengers affected.

• Disembarked at approx.. 4am and ushered to customer service desk. ONE staff member available for quite some time until several customers complained.

• Waited for 2 hours to be allocated a hotel and onward flight (and we were quite near the front of the line)– Virgin from JFK at 6.45pm – NOT United flight promised. No additional refreshments offered. Nobody knew what was happening.

• Given flight, hotel room until 1pm, $30 each of food vouchers, voucher for taxi to JFK.

• Told to collect baggage as needed to be taken with us due to airport change.

• Took baggage vouchers to reclaim at 6.15am. Told could wait up to an hour as baggage had to be located.

• Enquired several times and told ‘it’s a long process’. At approx.. 8am, all baggage from flight was unloaded after insistence of speaking to a manager and being told ‘it wouldn’t make any difference’ and refused.

• Bags collected approx. 8.30am.

• Shuttle taken to hotel and checked in approx.. 9.15am.

• Had 3 hours of sleep. Checked out due to need to travel to JFK. Used $30 of vouchers at hotel to buy food.

• Shuttle back to airport and then found cab to JFK.

• Arrived at JFK at approx.. 3.30pm and checked into Virgin flight.

• Tried to used remainder of food vouchers but told only valid at Newark by customer services. Therefore purchased own food and drink.

• Virgin flight left early and arrived early.

• Arrived back in UK approx. 20 hours late.

The whole incident was a complete shambles - very little communication, and what little we had seemed dubious. The customer service agents left to deal with the mess after the flight crew departed were not filled in on the situation and the extra 6pm flight promised by the ground crew did not exist.

What I'm looking for is some kind of idea what compensation we would be entitled to or could reasonably demand. If this were an EU airline, or a flight within the EU we would be entitled to several hundred GBP in compensation and a refund on our tickets, but I can't find any laws implying that is the case for US airlines. I'm thinking the bare minimum we should be entitled to is a full refund on the flight plus our out of pocket expenses.

Thank you for reading and any advice you may be able to provide.