Britain is likely to experience a surge in immigration amid the uncertainty of the negotiations to leave the EU, according to a committee of MPs, who say it is imperative for the government to offer EU citizens living in the country confirmation of their status.

The home affairs select committee called on ministers to prepare and fully resource the country’s immigration agencies to make sure they can handle the “heavy extra demands that will be placed on them by the fallout from Brexit”.

It also claimed it would become more difficult to remove offenders who are EU nationals from the UK “on paper” and in practice after Britain leaves the EU.

The committee chair, Labour MP Keith Vaz, said: “The biggest issue relating to Brexit is migration. There is a clear lack of certainty in the government’s approach to the position of EU migrants resident in the UK and British citizens living in the EU. Neither should be used as pawns in a complicated chess game which has not even begun.”

Facebook Twitter Pinterest The Labour MP Keith Vaz called for an end to the uncertainty surrounding the status of EU nationals. Photograph: Peter Byrne/PA

His report sets out three potential cut-off dates that the government could impose for new immigration rules to come into play: 23 June when the referendum was held, the date that article 50 is triggered, or when the UK actually leaves the EU.

“Unless the government makes a decision, the prospect of a ‘surge’ in immigration will increase. Multiple voices and opinions from government ministers causes uncertainty, and must stop,” added Vaz.

The report suggested there could be a sudden increase in migration from outside the EU due to anticipation of stricter rules being enacted to respond to national concerns about the numbers of people coming into Britain.

It also identified myriad other problems around people smuggling, failure to deport foreign offenders, and serious inefficiencies in the handling of migration data. The report found that:

One third of lorries – amounting to 750,000 vehicles a year – were arriving at the UK border without the recommended levels of security.

Truck drivers and companies were served 3,300 civil penalties for people smuggling, whether intentional or not, but £2.5m out of the £6.5m owed was never paid.

The number of foreign national offenders in the community “remains unacceptably high”, with the government falling short of a commitment to remove Polish offenders in British jails.

The Home Office has failed to put visa information on to computers and “lacks an effective and efficient system for managing its immigration casework”.

An “unacceptably high number of asylum applications are being dealt with inappropriately”, with the government returning people to unsafe countries such as Eritrea.

But MPs said Brexit was the top issue. The cross-party group concluded that the decision to leave the EU had left European citizens in Britain and UK nationals abroad in a “potentially very difficult and uncertain position”, which was why they wanted a cut-off date.

“EU citizens settled in the UK before the chosen date should be afforded the right to permanent residence. The challenge of successfully resolving the practicalities of the UK exit in relation to EU citizens must not be underestimated. A unit should be established in the Home Office to deal with this issue, in addition to the newly established department for exiting the European Union headed by the newly appointed secretary of state, Rt Hon David Davis,” the report said.

It suggested that the priority should be establishing where EU citizens live and work, with one option being to set up a system of registration.

A government spokeswoman said: “We have been clear that we want to protect the status of EU nationals already living here, and the only circumstances in which that wouldn’t be possible is if British citizens’ rights in European member states were not protected in return,” she said.

“We are about to begin these negotiations and it would be wrong to set out further unilateral positions in advance. But there is clearly no mandate for accepting the free movement of people as it has existed up until now.”

Responding to other parts of the report, a spokeswoman for the Home Office said it was in their interest to combat any attempts to enter the UK illegally, not least because it hit the haulage industry and put drivers’ safety at risk. They said they had worked closely with companies to set standards.

On money owed to the government she added: “Any outstanding fines will be aggressively pursued and we will detain the vehicles of those concerned if necessary.” The department suggested that it was misleading to give a figure of unpaid fines as money was collected in the following financial year.

The spokeswoman said the government was determined to act against employers who use illegal workers, would consider the findings on processing data, and added: “Foreign nationals who abuse our hospitality by committing crimes in the UK should be in no doubt of our determination to deport them.”