The home secretary, Amber Rudd, has been warned by a cross-party group of MEPs that her plans to force EU nationals to add their names to a register in the transition period immediately after Brexit would be illegal and unacceptable to the European parliament.

The MEPs from across Europe have written to Rudd following her suggestions to the home affairs committee that she would expect EU nationals to have to register with the authorities in the period immediately after Britain left the EU. Brussels is planning to insist that a transition period after the UK leaves in March 2019 would involve Britain remaining under EU law and all its institutions, without exception.

The MEPs wrote: “Is the Home Office suggesting that only non-UK EU citizens needs to register? Article 26 of the freedom of movement directive makes it very clear that residency cards are for everyone, or no one.

“We find it extremely troubling for the home secretary of a member state currently complying with EU laws to make such a statement.”

The MEPs – Sophie in ’t Veld, Seb Dance, Jean Lambert, Claude Moraes, Beatriz Becerra, Cecilia Wikström and Catherine Bearder – said: “We also heard you say during the committee hearing that those EU citizens who fail criminal records checks may be rejected.

“Article 7 of the freedom of movement directive clearly states the necessary areas of compliance for an EU citizen to reside in a member state and criminality is not one of them.”

The European parliament will have a veto on any withdrawal agreement between the UK and the EU, including the terms of a transition period. While some member states do insist on EU nationals registering, their laws do not discriminate between their own citizens, those from other EU member states or non-EU nationals.

The warning came as figures released under a freedom of information request showed the Home Office was struggling to cope with the mountain of applications from EU citizens for permanent residency documents, with waiting times for some submissions tripling.

The request by Colin Yeo, an immigration barrister, discovered the average processing time had increased since 2015 for all types of documentation available to European Economic Area citizens.

Permanent residency certificates – which can be issued to an EU citizen or their spouse during their initial five years in the UK – take an average of 116 days to obtain, according to the latest figures released by the Home Office, from the fourth quarter of 2016. This is three times longer than the average for the whole of 2015, when it took just 43 days to get a certificate.

Permanent residency cards take even longer, with figures for the third and fourth quarters of 2016 showing it took an average of 170 days to get a reply from the Home Office, compared with 125 days in 2015.

The Home Office resisted releasing the figures when first requested by Yeo. It delayed release for eight months and only did so after an official warning from the Information Commissioner.

EU citizens are not required at present to have any paperwork proving their right to live in Britain, but many were panicked into applying for permanent residency documents and certificates after the referendum last year.

Nearly 30,000 EU nationals applied to become British citizens in the 12 months after the Brexit vote on 23 June, almost double the number in the previous year.

Home Office statistics show 28,502 applications were submitted between July 2016 and June 2017, up from 15,871 in 2015-16.

On Tuesday, Rudd revealed that an extra 1,200 staff were to be recruited by next April in order to establish an “easy access” registration process for EU nationals.

The home secretary said those currently resident in Britain would be “able to regularise their position” by applying for settled status biometric residence permits if they had lived in Britain for more than five years. Those who failed criminal record checks or identity checks could be rejected.

The Treasury has made £50m available to fund the preparation costs this financial year for the new registration system.

However, some of the largest employers in London, including banks, supermarkets and hotel chains, have told the Guardian of serious reservations about the capacity of the British government to establish such a system.

Mark Hilton, the employment and immigration director at London First, whose members include household names such as Marks & Spencer, HSBC and Chelsea football club, said: “Whilst we welcome May’s intentions to streamline the application process, there’s little to assure us on how this will happen. We’ve seen how so-called streamlining to the welfare system has played out with universal credit. We need much greater confidence that this new system will work.”

• This article was amended on 24 October 2017. An earlier version referred to the home secretary saying that those who were “unemployed” could be rejected for settled status biometric residence permits. This has been corrected.

