We won't be flying in or out of Heathrow ever again. That decision is on past experience over the last 10 years - not just the chaos of today. Ask your agent for a "cruise only" price and trawl through the airlines from your local airport on tinternet.
My last princess cruise the flight was provided by Titan airways no problems at al.
Went to the Caribbean with Princess - BA to Gatwick and Thomsonfly to San Juan, Puerto Rico. Going with them again - Central America and Panama Canal. British Midland to Heathrow, American Airlines Heathrow to Los Angles and return and then BA Heathrow to Manchester. Hope you enjpy your cruise.