The union claims a backlog of up to a million undelivered items has built up
Claims that "mountains of undelivered mail" are stuck in Swindon are "completely misleading", according to Royal Mail.
The Communication Workers Union (CWU) said it was due to three 24-hour strikes over the past fortnight during which more than 800 workers walked out.
Further industrial action in the coming months has not been ruled out.
But the CWU said some progress had been made in locally organised talks with Royal Mail.
For its part, Royal Mail said 90% of staff were working normally across the UK.
CWU official Kevin Beazer said: "There is a hell of a backlog of up to a million items - in trailers, at Reading, and Oxford, and throughout the south west where action took place last week. Offices are stuffed with mail.
"We talk to postmen and women in delivery offices every day and they said they are fed up with complaints from customers and businesses.
"They now hand out pieces of paper with the office manager's number on to ring up and complain."
A spokesman for Royal Mail said: "Whilst it is obvious that some customers will suffer disruption to their mail service as a result of the CWU's damaging strike action, the claims about the operation in Swindon are completely misleading and are in no way a true reflection of the situation.
"We do not have mail stored in vehicles, this is simply scaremongering."