Are the dates and times shown for the USA only or is it possible to get the information that is relevant to the UK?
It's for the country/network a show premieres on.
If a show premieres in the USA, you can't get information on UK dates.
If it premieres in the UK, there won't be any USA dates.
Dates for other countries is a pending feature, you can vote for it here:
https://www.tvmaze.com/threads/128/support-for-premiere-airdates-in-different-countries
Check the calendar, it should show local times relevant for you.
Unless you want the show times for your local networks. That will hopefully never happen as this is not a page for tracking network schedules.