Whether the traveler could transit airside seems to be irrelevant. A Turkish citizen holding either a valid D visa or residence permit issued by an EEA state (e.g. Germany) meets to requirements to enter the UK under landside transit without visa rules. So, the traveller would have been able to enter the UK as long as their onward flight was scheduled to depart by the next day. The rules are available here: https://assets.publishing.service.gov.uk/media/65ce4119e1bdec0011322213/UK+Visa+requirements+February+2024.pdf
A query on Traveldoc also confirms this:
Direct Airside Transit Visa (DATV) is not required for passengers holding a valid Common Resident Format Residence Permit issued by a European Economic Area member state or Switzerland, when transiting Gatwick Airport (LGW), London Heathrow Airport (LHR) or Manchester Airport (MAN) and the passenger's next destination is not Ireland. Passenger must hold an onward flight, correct documentation for their destination and depart the same calendar day. E-visas or e-residence permits are not acceptable unless the airline is able to verify it with the issuing country.
Visa is required for passengers who leave the international transit area of the airport to transit land-side, or otherwise have to pass through border control.
This does not apply to passengers holding a confirmed onward ticket to depart the United Kingdom by air before the end of the next calendar day, and one of the following exemption documents (electronic visas or residence permits are not accepted) provided complying with any conditions specified:
[...]
- Valid category "D" visa or common format residence permit issued by a European Economic Area member state or Switzerland;
I do not know the exact procedures for claiming compensation in this case for you. Maybe someone else can shine light on this.