Campaigners for EU citizens have written to Theresa May demanding a lifetime guarantee of existing rights after Brexit to “break the deadlock” in talks.

In a strongly worded letter, British in Europe warns May that if she jeopardises the rights of British nationals settled in the bloc, negotiations will hit the buffers.

“If Michel Barnier, the European parliament and the EU 27 member states see that the UK government is willing to throw us, our rights, our livelihoods and our children under the Brexit bus, they will have no confidence that it will recognise and protect the rights of the 3 million or any other post-Brexit obligations in the future,” the campaign group, which represents 11 citizens groups across the EU, said.

They urge the prime minister to “stop treating citizens’ rights as a minor issue “that can be pushed aside in the headlong dash for a trade or indeed a security agreement” and plead with her to end the “platitudes” about their contribution to national life.

In reference to the UK’s offer, they also implore her to “please stop treating citizens’ rights as an immigration issue”.

With UK and EU negotiators due to start the fourth round of Brexit talks next week, the British in Europe lobby group says the uncertainty about their futures will hang over them like a cloud for life unless the prime minister delivers on her Florence speech promise that “you can carry on living your lives as before”.

“We don’t share the PM’s view that significant progress has been made on this most fundamental of issues and are thus astonished that our prime minister missed such an important opportunity to remedy this, not just for the 4.2 million citizens affected, but for the future of the UK-EU relationship post-Brexit,” said Jane Golding, chair of British in Europe. The group is a coalition of 11 campaign organisations representing the 1.2 million Britons on the continent and the 3 million EU citizens in the UK.

“We now appeal to her to listen to the voice of British citizens in the EU and all those EU citizens who make such a valuable contribution to the UK,” she said.

The coalition is sending what it says is “a comprehensive letter” to May asking her to scrap proposals to make all EU workers and their family members in the UK to apply for a new “settled status”, a proposed new category of immigrants.

EU citizens argue they settled in other countries in good faith under EU freedom of movement laws and that Britain should not renege on these in the case of the already settled populations just because of Brexit.

They want her to return to her Lancaster House promise to ensure the status quo for EU citizens in the UK in order to get a mirror deal for Britons in Europe.

The letter will say that the repercussions on the lives of British working families and pensioners in Europe could be “disastrous”, as this could be reciprocated by any country in the bloc in retaliation against the UK.

“Her speech gives no clear comfort that she has listened to us and is moving in the right direction,” Golding said.

The group wants Britons who have settled in countries on the continent to continue to have freedom of movement, while EU citizens in the UK want the government to abandon its offer of “settled status”.

This is important, they say, because it will give them the lifelong right to return to the UK if they are forced into taking out foreign citizenship because of Brexit.

More immediately, it will allow them to continue to do cross-border business on the continent. Immigration barrister Jan Doerfel says May’s fresh remarks on EU citizens “fall far short” of the expectation of the EU.

“Theresa May’s speech does not bridge the substantial gap between the EU and UK positions and does little to advance progress on the issue of EU citizens’ rights,” said Doerfel, who is concerned particularly in relation to the future of family reunion rights.

“While May has now agreed with the EU position that future case law of the court of justice of the European Union can be taken into account, she has not agreed to CJEU oversight.

“Her voiced commitments hence fall far short of the basic EU position as summarised by Michel Barnier in response to Ms May’s speech that ‘EU27 citizens in the UK must have the same rights as British citizens today in the EU’ and that these rights ‘must be implemented effectively and safeguarded in the same way in the UK as in the EU’,” he said.