The pop superstar is playing stadium shows in Cardiff, Edinburgh, Sunderland and London in May and June.
Some users had problems with the Ticketmaster website, while others said they were kicked out of the queue, which in some cases was 500,000 strong.
Three more London dates were added on Tuesday "due to high fan demand".
A Ticketmaster spokesperson said: "We experienced and successfully handled an extraordinary level of demand and traffic today for Beyoncé.
"While there were never going to be enough tickets to meet demand, thousands of happy fans secured their tickets."
A portion of the UK tickets had already been sold in a series of pre-sales, while the first pre-sales for her North American dates began on Monday.
There is particular scrutiny in the US of how Ticketmaster handles the tour, especially since the company's systems were overwhelmed by demand for Taylor Swift tickets last year.