As we seem to be looking at the application of EC 261/2004-the flight appears to have been from Turkey to the UK.
Ec 261/2004 would only apply if the carrier was a "community carrier".
Is the carrier a community carrier? The carrier may have its "legal base" in Northern Cyprus-which is now part of the EU but for EC 261/2004 the determining criteria would be which state issued its Operating Certificate (according to the definitions within EC 261/2004).
This isnt clear-not least from the carriers own web portal. There are other suggestions that the airline's operating certificate might in fact be Turkish. (Whether the carrier would wish to make this an issue is anyone's guess if it came to an argument but it woul be the determining criterion).
Two points.
Art 19 of the Montreal Convention which would apply for air carriage between Turkey and the UK would ostensibly allow passengers to claim for damage resultant from delay during the course of carriage by air. This may apply a possible alternative cause of action. If the place of destination was England this would also allow an English court to be a forum for proceedings.
Did any of the family members also incur any other liquidaited and proveable damages such as loss of earnings? This might be recoverable under this possible claim?
(There may also be possible and potential contractual law remedies but the international aspects may make this more problematic).
Leaving aside the theoretical aspects-if the carrier chose to ignore any submitted claims-it might be worth looking at insurance policies for any legal expenses cover they provide. It appears only relatively few travel insurance policies provide legal expenses insurance for a dispute regarding the journey (as opposed to a claim for personal injuries-the type of claims that tend to get lawyers excited anyway). (There appear to be some exceptions such as Direct Line and Churchill)
Some household (house contents) insurance policies have wider coverage for legal expenses for generic consumer disputes that might include such a potential dispute with an airline. It would be worth checking if you are serious in taking matters further.
Finally with regard to the travel insurance "late departure" compensation amounts.
Here is a potentially interesting point ( I think). Could your family members claim these amounts and also make an additional claim against the carrier for their losses?
If the policy is not one of indemnity (which a fixed tariff of compensation would suggest it isnt) does the general rule of subrogation apply? If the travel insurance policy does not apply a specific assignement to the insurer in the policy wording-is there anything precluding the passenger claiming from the travel insurer for their fixed "compensation" and also claiming from the airline for liquidated losses?
This may be an interesting question to ask the travel insurer if a claim is made against them in any event. (i.e does the travel insurer assert a right of subrogation or assignment-and if so why and how?)
There is a specialised discussion forum at Flightmole that looks in detail at claims under EC 261/2004/Montreal Convention-where many people have successfully challenged airlines under EC 261/2004 and obtained compensation and looks at these claims in more depth. May be worth a look.