Last week, the fried chicken chain switched its delivery contract to DHL, which blamed "operational issues" for the supply disruption.
It was unclear when the delivery problems would be rectified, a KFC spokesperson said.
Workers are being encouraged to take holiday but would not be forced to do so, he added.
"Our teams are working flat out all hours to get the rest back up and running as soon as possible - but it's too early to say how long it will take to clear the backlog."
KFC has set up a web page where "fans" can find their nearest outlet that is still open.
It said that in the restaurants owned by the chain, staff on short-term contracts would be paid the average hours worked per day over the past 12 weeks, while those on salaries would be paid as normal.
However, 80% of KFC outlets are run on a franchise basis.
"Franchisees will be seeking their own independent advice, but we're encouraging them to adopt this policy too," said the chain.
Until last Tuesday, KFC's chicken was delivered by South African-owned distribution group Bidvest, which describes itself as "the leading supplier of logistical and supply chain solutions to the UK hospitality and restaurant sector".
But after the change in the contract, many of the food giant's outlets began running out of chicken products
