The pop superstar is playing stadium shows in Cardiff, Edinburgh, Sunderland and London in May and June.
Some users got an error message on 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”, taking the number of UK dates to eight.
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.
The UK concerts are part of a world tour in support of her Grammy-nominated Renaissance album.
The shows kick off in Sweden on 10 May, before landing at Cardiff’s Principality Stadium a week later.
She will then visit Edinburgh’s Murrayfield on 20 May, Sunderland’s Stadium of Light on 23 May and London’s Tottenham Hotspur Stadium on 29 and 30 May.
After tickets went on general sale on Tuesday, she added three further shows at the Tottenham stadium on 1, 3 and 4 June.
In the UK, standard ticket prices to see Beyoncé range from £56 to £252.
However, the VIP packages, some of which include guaranteed front row seats or even seats on the stage, stretch to more than £2,300.
Many fans on social media ridiculed the cost, reminding Beyoncé that she advised people to “quit their jobs” on her recent single, Break My Soul.
Billboard magazine predicted the five-month tour could earn the star up to $275m (£223m).
Ticketmaster said it had changed its process for the North American legs of Beyoncé’s tour following the Taylor Swift debacle in November.
The company apologised after bots disrupted the sales for Swift’s Eras tour. Ticketmaster said there was a “terrible consumer experience”, while Swift said she was annoyed that many fans felt “like they went through several bear attacks” to get seats.
The US Senate is now investigating the company.
Beyoncé fans were asked to register for Ticketmaster’s Verified Fan process – which the company says filters out touts – before the first North American pre-sale opened on Monday.
Those who managed to sign up ahead of time were entered into a “lottery-style process” after demand outstripped the number of available tickets.
If there are remaining tickets after those on the waiting list have been offered a chance to buy, then they will go on general sale – but that is unlikely.
Source: BBC NEWS.com