"Although immigration concerns have dropped, Leavers are starting to worry more about everyday matters, particularly rising utility bills and food prices," said Steve Smith, Nielsen's managing director for the United Kingdom and Ireland.
Many voters who backed Leave come from poorer households which are likely to face the biggest pinch from rising prices for food and fuel.
The BCC said raw material costs were having the biggest influence on prices since late 2011, and called inflation "a key risk to the UK's growth prospects".
Official data on Tuesday showed consumer price inflation held at its highest in nearly four years at 2.3 percent in March, while the pace of factory-gate inflation barely eased from February's five-year high of 3.7 percent.
Nonetheless, confidence at services firms and manufacturers was its highest since the first half of 2016, the BCC said.
Services companies, which make up the overwhelming majority of Britain's private-sector economy, reported the fastest growth in domestic and overseas sales since June's referendum.
Manufacturers enjoyed the most rapid domestic sales growth since the third quarter of 2015, while export sales grew at the quickest pace since the final three months of 2014.
The BCC survey was based on responses from more than 7,300 firms polled between Feb. 20 and March 13. The FSB study was based on 1,245 firms contacted in January and February.
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