If the little Princess Elsa living under your roof is driving you mad with the Frozen soundtrack on loop, then you may be one of the country's legions of parents scouring websites and toyshops to satisfy demands for a mini-me outfit.

Fear not, because another shipment of polyester Disney Princess chic will be landing on British shores soon after Disney asked suppliers to step up production of the clothing, dolls and soft toys based on the blockbuster animated musical.

For months parents hunting for Elsa's shimmering blue dress or Anna's caped ensemble and plaited wig have been left frustrated by empty shelves as Frozen-mania grips Britain, with Disney admitting that the industry had underestimated the film's success.

On the Mumsnet website, members regularly swap intelligence on how to track down the coveted "Elsa dress" – in the face of profiteering on eBay, where the outfit can sell for more than £80. "I've found Frozen stuff a nightmare to get hold of," complained one member. "The dolls are in and out of stock everywhere and vary massively in price. The only clothing I could find was a T-shirt in Tesco that I bought in desperation in the end!"

Frozen, based on Hans Christian Andersen's fairy tale The Snow Queen, is officially a blockbuster franchise after it overtook Toy Story 3 in March to become the highest grossing animated film of all time, raking in $1bn (£644.5m) at the box office. It was firmly lodged in the UK box office top 10 for more than three months and now ranks as the 10th highest grossing film in cinema history.

The success belied an inauspicious start in 2013 when its release met a lukewarm critical response, one reviewer bemoaning – contrary to many households' experience – the lack of "memorable tunes". But what did they know? Children, with their parents in tow, have voted with their feet over and over again. The film won two Oscars, and its songs, penned by Broadway duo Robert Lopez and Kristen Anderson-Lopez, topped the US Billboard 200 chart for 13 consecutive weeks. A Broadway musical is now in development.

After the recent release of the DVD, Frozen mania is spreading still further, with five-hour queues reported for the chance to meet the Elsa and Anna characters at Disney theme parks. It is even a hair phenomenon, with YouTube videos demonstrating how to create Elsa's French braid getting millions of hits.

London-based Sachaa Pattison, who appears as Elsa at Frozen-themed kids parties, said the film was popular with boys as well as girls. "I've got sons of four and seven and they are both obsessed and sing the song Let it Go in the playground at school," she says. "When you hear Frozen it is like a Broadway musical." Pattison said the party company she works with has been inundated with requests from mums who have booked Disney Princess themed parties and want to change the character from Rapunzel or Cinderella to Elsa or Anna.

Faced with Frozen's popularity and disappointed children, Disney imposed quotas on some of themerchandise so as to create an "equitable way" to buy the most popular stuffed animals and dolls. Shoppers on the Disney website can put only one of the £15 Elsa and Anna dolls in their basket. Some of Disney's US stores have even been running lotteries on a Saturday morning with the winner able to buy the prized Elsa costume. The company does not expect stock levels to return to normal until the summer. "The popularity of Frozen has been phenomenal," said a Disney UK spokesperson. "We are thrilled that audiences have connected with the characters and story in the way that they have and we are working hard to keep bringing additional product into stores as quickly as possible."

Trying to explain the merchandise shortage to Wall Street last week, Disney's finance director, Jay Rasulo, told analysts that toy manufacturers had to "hedge their bets" when a new film came out. Notable flops of recent years have included the 3D animation Mars Needs Moms. Frozen's success caught everyone off guard, with Disney stores already selling double the amount of related paraphernalia as for the 2010 princess movie Tangled.

Experts have likened the sales sensation to the Cabbage Patch Kids craze in the 1980s, with Sean McGowan, a toy industry analyst at Needham & Co, telling the New York Post: "We're now at the stage where the demand is almost being driven by the scarcity because of the social status attached to being able to find it."