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UK is outside the Schengen area. So, this itenary shouldn't be an issue as long as you stay within the international transit area of the airports.
In detail:
If you had selected a flight that has layovers in two Schengen countries, it would have been an issue, as Schengen-Schengen flights is like a domestic flight and you have to clear immigration, which require a Schengen visa. Since one layover is in UK, which is outside Schengen area, you can stay in the international transit area and you don't need transit visas.