A flood of applicants for renewed or first-time passports meant there were no one-day appointments available for up to a fortnight.
Others were told by the Identity and Passport Service that the only slots available were miles away from where they lived, according to the Daily Mail.
Paul Jessop, 52, from Petersfield in Hampshire, told the newspaper the earliest appointment he was offered was in Durham August 4.
He had contacted the IPS because he wanted to get a passport for his six-year-old daughter Rose so she could travel with him to Spain to visit his sister.
Mr Jessop said: "I rang them this afternoon and after about 10 minutes of waiting they said the first appointment they could offer was on August 4. It's insane.
"They could do nothing in less than three weeks despite saying there's one-day appointments. The one-day emergency appointment is nothing of the sort. I was lost for words."
When approached for an appointment to replace a lost passport, the IPS only had appointments in a fortnight for anywhere in England and Wales other than Durham.
A spokesman for the Public and Commercial Services Union said: "The chaos we've all seen at the borders recently is mirrored in the passport agency, where bungling ministers have cut so many jobs they can't cope with the demand.
"It's sheer incompetence, and we've been warning for months that this would happen. It's not difficult to work out that during the summer lots of people will want a passport."
A spokesman for IPS said: "The peak period for passport applications is under way, and we are reminding customers to leave at least three weeks for a renewed passport to arrive and six weeks if applying for the first time.
"We are successfully issuing 99 per cent of renewed passport within three weeks. We are also successfully issuing 97 per cent of first-time passports within six weeks.
"We may not always be able to offer an appointment on the same or next day.
"However, cancellations do occur and people can contact our customer service team to see if any are available at short notice."