Generally speaking, from reports etc over the years, and if you are subject to England and Wales school holidays (as Scotland and Ireland are different), Summer is busiest, followed by October half term if its the same week as most European schools (ie this year, when October half term is w/c 29th October), then Easter (as again, most Europeans have Easter weekend itself as school/office/public holidays).
May half term beginning of the week can be busy if it includes Whitsuntide, because again, many European countries have Whit as a public holiday. This year, our June half term should be fairly quiet in comparison, because there are no public holidays/school holidays except in the UK that week

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The quietest English/Welsh school holiday of the year is generally February half term, as usually there are no other European school holidays at the same time (but you'd need to check in advance of booking to make sure). However, its also by far the coldest time of year to go, and often there are major refurbs going on in the parks in the period between January to March each year.