I second this. I can't imagine that there will be any impact for the last leg of your ticket, IST-MNL if you have TK flight to IST during your layover. On which basis? In principle, you could be returning from LHR within the 2 days layover and continue your journey.

In fact I think there is a very good chance to persuade the check-in agent at BOG to check your bags to LHR if you explain at check-in that your destination is LHR rather than IST (and show the IST-LHR ticket). Again, why not?- you are not doing anything wrong let alone illegal. TK has done this for me in the past without issues.