Barrowload turns out to have strong regional variations. In the News on the Web corpus, there are 24 instances of the word in British news reports and a few from Ireland and New Zealand, but absolutely none from the US, Canada or Australia.
The word is mostly used figuratively, for a large quantity. "They went on to dominate the second half, creating a barrowload of opportunities" (UK newspaper the Courier). Or "books that sell by the barrowload" (Bob Geldof).
The word is not listed in any of the dictionaries I consulted, apart from Wiktionary. However, adding -load to a word for a means of transport is a fairly standard construction. We already allow many such compounds: boatload, busload, carload, cartload, etc.
The word is not completely alien to North American writers, but generally used in a literal sense. The citation given by Wiktionary is from Canadian writer Emily Carr: "millions of barrowloads of earth and rocks". And US crime fiction writer Donald Westlake used the word in one of the stories in his collection Thieves' Dozen: "In a Millet over the mantel, a French farmer of the last century endlessly pushed his barrowload of hay through a narrow barn door."
There's no way the word could be accepted as common, given its rarity in American, and Australian, sources, not to mention its absence from almost every dictionary. However I'll accept it as a rare word.