score:4
To close this loop, I am answering my own question.
It's been 6 weeks after submitting the original claim. In this time, I had to chase it once via their website "feedback" section, then via their facebook page, then I had to submit the claim again, because they stated that they never received it (note that I had all submission reference numbers that I always provided to them).
Now, about 3 weeks after the second time I submitted it, I finally got an email confirming partial refund. As expected, they don't refund Β£400, because the flight is on a non-EU airline and is not originating from the EU. They also do not refund/reimburse for any expenses incurred after reaching the final destination, regardless of whether they were directly incurred as a result of the delay.
Overall, I received airline voucher compensation for the visa and food expenses and the letter states that I can exchange this voucher for cash at any of their sales offices, which I will be doing promptly. The compensation is in Turkish Liras, even though some of my original expenses were in GBP.
So far this leaves me about Β£25 out of pocket and half a day of (paid) holiday leave at work. I could probably pursue them in small claims court for the remaining costs, however the time I would spend on it would be worth more than the amount I could recover, so I'll leave it it at that.
EDIT: To answer the question that was asked later, yes, I did exchange the vouchers for cash at Heathrow ticket office, as I was planning to.