An interesting question indeed. A non-EU airline can fly from the UK outside the EU if it has an established operation in the UK and if the foreign destination country chooses to accept these arrangements.
There is not a hope in hell of the Turkish authorities accepting a Swedish carrier to fly Gatwick-Dalaman. They have always protected their home airlines (more than UK PLC has done!) and will continue to do so. The Egyptians might let a Swedish airline fly Gatwick-Sharm if they are given enough cash to make them complicit.
If Kiss want to fly their full programme then they will either need to lease the aircraft into a UK AOC holder or get a UK AOC full stop. Rumours of Kiss buying the Markoss operation at Biggin Hill are probably not misplaced - it would be a useful flag of convenience.
The denials by Kiss of involvement by PW are just unbelievable. 99% of industry people will tell you that PW is involved if you did a quick survey to establish their views. No one, but no one, believes the stories being put about that he's not behind it. Two ex-XL people at Meridian Aviation is company but six is a crowd. As for the spelling mistakes - well, it was always ironic that the Icelandic people at XL could spell and use punctuation marks far more competently than the Brits.