Strict campus lockdowns and a worsening mental health crisis are creating a “perfect storm”, student leaders have warned, after tensions boiled over in Manchester where fencing erected around accommodation was pulled down.

Students at other campuses are organising rent strikes and complaining of a lack of mental health support against the backdrop of England’s new national coronavirus lockdown, under which they have been told not to return to their family homes during term time.

In cities such as Nottingham and Lincoln, there has also been a backlash among students and staff towards universities, which are insisting that students and lecturers travel to campuses for face-to-face teaching.

Student leaders in Leeds have levelled accusations of heavy handedness against security guards on campuses elsewhere, including the use of dogs to patrol halls of residences.

Cambridge University was accused of sending stern enforcement emails warning that any students who returned home during term-time for extended periods would fail to graduate or progress to their next year, unless they had an exemption on health grounds. While the policy pre-dates the pandemic, students had hoped it would be retracted.

Ben Margolis, the undergraduate president of Cambridge students’ union, said colleges had been applying the rules inconsistently, with some more lenient than others. He added that some students had left without permission, while international students were particularly angry.

Meanwhile, a day after fences aimed at controlling the spread of coronavirus were torn down around halls of residence on the Fallowfield campus in Manchester, students awoke to early-morning fire alarms and evacuations.

“We all thought it was slightly suspicious,” said George Rogers, a first-year student. “Fire alarms are usually Wednesday between 11am and 12pm and last 10 seconds, this was a full fire alarm. It was so cold.”

The university said these were scheduled fire evacuation practices, in line with the UK code of conduct, that were postponed from the start of term.

The university’s vice-chancellor Dame Nancy Rothwell also announced plans to launch an inquiry into the decision to erect the fences, and the “very poor communication” surrounding it, with a report to be finalised by the end of November. She said: “I regret that our actions led to protests and unrest at the residences last night.”

On Thursday night, hundreds of students attended the protest to tear down the fences. The demonstration was organised by individual students along with Rent Strike Manchester, a student collective campaigning for a 40% rent reduction and better support for students by withholding rent payments on their halls of residence.

Molly, an organiser and first-year economics student, said the protest had galvanised students, some of whom now plan to take part in the rent strike.

“We realised people were more angry and frustrated than we thought initially, so there’s a good sense of solidarity and people are excited to continue to take action,” she said.

Rent Strike Manchester said its WhatsApp group chat had increased by approximately 150, its Twitter account went from 300 follows to 1,200 and its Instagram page gained 1,000 followers.

Students at other universities, including Glasgow and Liverpool, are planning to hold solidarity marches with Manchester next week, organised by grassroots groups under the name Students Before Profit, which is campaigning for better support.

Students at a number of universities are unhappy with the level of mental health support they have received, with complaints about services that are hard to access, with long waiting lists or limited to 15-minute time slots.

Amy, a student at Reading University, said she had struggled to get an appointment with a counsellor. “I’ve never had any real mental health problems before Covid-19, but in the last six months I don’t think I’ve ever been so low, anxious, stressed and depressed,” she said.

Leila Malik, a second-year student in Manchester who set up a Facebook group to compile stories from students about their experiences of anxiety and depression, said: “The workload and academic pressure is the same, if not more, than previous years but we can’t perform as normal because we’re not getting the same education.”

The National Union of Students is campaigning for students to have their fees reimbursed if they want to leave their courses. “We’re seeing what can only be described as a pressure cooker atmosphere on university campuses right now,” said Larissa Kennedy, the NUS president.

More than 1 million students who had been encouraged to move across the country during the pandemic arrived on campus only to realise that they had been “sold a lie” about the possibility of the possibility of face-to-face teaching because their institutions were preoccupied with collecting fees and rent, she added.

A Cambridge spokesperson said: “The University of Cambridge is following UK government guidance given to all universities in keeping the university and colleges open, and in asking students in residence to remain here and avoid travel while the national lockdown is in place.”

A University of Leeds spokesperson said: “Dog patrols are a familiar sight on university property, including around halls of residence, and are there to serve as a visible source of reassurance to staff and students.

“We are unaware of any complaints about the dogs, and our students have always been generally reassured by their presence.”

• This article was amended on 9 November 2020 to remove some personal information.