The British-English expression trolley-dash denotes:
1-: An event in which the winner of a game or competition is entitled to a set period of free shopping in a supermarket or other store, the object being to place as many products as possible in a shopping trolley during that time.
2-: A quick or rushed shopping trip around a supermarket or other store.

In this expression, the noun trolley denotes a large rectangular four-wheeled metal basket provided for supermarket customers in which they may collect goods for purchase.

Note: The noun trolley has a different meaning in the phrase off one’s trolley.

These are, in chronological order, the earliest occurrences of the expression trolley-dash that I have found:

Sense 1: An event in which the winner of a game or competition is entitled to a set period of free shopping in a supermarket or other store, the object being to place as many products as possible in a shopping trolley during that time:

1.1-: From Shopping dream prize, published in the Atherstone Herald (Atherstone, Warwickshire, England) of Friday 18^th March 1977:

A housewife’s shopping dream—that was the top prize in a Spring draw at Tamworth’s Mercian School at the weekend.
The prize gives a lucky lady shopper the chance to take her pick from the shelves of a town supermarket without having to pay a penny for the goods.
But perhaps the main drawback to the money-saving venture, aptly called a trolley-dash, is that the shopper gets only two minutes to make her selections.
The fast-moving feat takes place next Thursday by courtesy of Tesco supermarket and the Mercian School’s Parents’ Association.
The Association is putting £50 to cover the prize and Tesco’s will meet the difference if the shopper collects more than that amount in goods.

1.2-: From Elsie was prepared!, published in the Evening Chronicle (Newcastle upon Tyne, Tyne and Wear, England) of Wednesday 23^rd December 1981:

Housewife Elsie Ramsay took a tip from the Scouts when she won two minutes free shopping in a superstore.
Elsie, aged 45, immediately went to the SavaCentre store in The Galleries, Washington, to prepare.
She and her husband, Don, an electrical engineer, worked out a plan of action and a route for the trolley dash.
And it paid off. The mother-of-four filled three trolleys with goods worth £118.51.
Her prize was the first in an annual Christmas draw, organised by Washington Biddick Farm Arts Centre.
Elsie, of Manor Park, Concord, Washington, said: “I still had some shopping to do for Christmas when I was told I had won and it came in very handy.”

1.3-: From the Evening Post (Reading, Berkshire, England) of Friday 26^th February 1982:

Dash of luck

Lucky Mrs Gloria Fraser went on the shopping spree of a lifetime yesterday.
She packed two trolleys full of goodies—and didn’t have to pay a penny for it.
But it was no ordinary shopping trip for the 40-year-old auxiliary nurse of Haddon Road, Maidenhead. She had just two minutes to fill her trolleys.
Mrs Fraser (pictured) won the high speed trolley dash round SavaCentre at Calcot, Reading, in a raffle.
The dash was confined to the foods section of the store and she was allowed only one of each item. But she rang up an impressive bill of £233.93.
Before the dash Mrs Fraser said: “I’m just doing it for the fun of it. I feel embarrassed about it because it seems so greedy.”
Two minutes and two full trolleys later she was exhausted. “I feel weak at the knees,” she said.
And her husband Bill, aged 52, joked: “She doesn’t normally go round at that speed—we couldn’t afford it if she did!”
Mrs Fraser said she would be giving some food to her mother but most of it would soon be eaten by her three sons. She added she hoped to make up a parcel of food to be raffled in aid of the Guide Dogs for the Blind Association.

David Prentice used trolley dash figuratively in his column, The Final Word, published in the Liverpool Echo (Liverpool, Merseyside, England) of Monday 22^nd November 1993—Graeme Souness (born 1933) was at that time the manager of Liverpool Football Club; Harrods is a prestigious department store in London, England:

Souness has been allowed to buy like a down-and-out on a trolley dash in Harrods.

Sense 2: A quick or rushed shopping trip around a supermarket or other store:

2.1-: From Brian Hill’s column, Holidays, published in the Liverpool Echo (Liverpool, Merseyside, England) of Tuesday 25^th January 1994:

New York is one of those places that can cost you the earth or be done on the cheap, depending on the width of your wallet. If you fancy taking in the odd Broadway show and a trolley dash in Tiffany’s take out an overdraft first.

2.2-: From Silent night at the tills, by Alex Duval Smith, published in The Guardian (London and Manchester, England) of Saturday 3^rd December 1994:

It’s Christmas shopping time […].
[…] Today has been deemed National No Shopping Day.
Not heard of the event? That’s only natural, as Enough!, the anti-consumerism campaign behind it, is not too fond of hype. “We are trying to encourage people to think twice about the peddled illusion that commercialism makes us happy,” says Paul Fitzgerald. […]
[…]
“People have forgotten the essential message of A Christmas Carol—to be nice to each other—and swapped it for a trolley-dash,” says Fitzgerald, a professional cartoonist.