I've cruised in UK region in summer and where I hit the really bad crowd problems were the international arrivals areas in airports. Once spent over 2 hours in passport line at Heathrow and almost missed my connecting flight. But in general things will be busier in those months which I why i prefer May and September. If your journey involves any trains, they will also be busy so it pays to book first class where available.
However, with fewer choices in rebooking all my cancelled 2020, the 2021 cruises, I've also ended up with a July/August cruise this year.