Pressure is mounting on private schools to open their land to the public and ease overcrowding in city parks, with a key government adviser urging ministers to requisition them to help avoid potential social unrest during the lockdown.

Prof Susan Michie, of the government’s scientific advisory group on coronavirus, said that private green spaces should be officially commandeered by the state to ensure everyone can exercise safely while maintaining social distancing.

Along with independent schools, Michie said private parkland and golf courses should also be temporarily used by the government to help city dwellers improve their physical and mental health and tackle frustration during the lockdown.

There are 130 independent private schools in Greater London alone and some have huge grounds, such as Harrow school, which has 300 acres – bigger than east London’s Victoria Park, which reopened on Saturday to large crowds and strict stewarding.

Across the UK, police said that the vast majority of the public had abided by the coronavirus lockdown rules and stayed at home for Easter, although officers in Essex stopped a number of motorists, whose excuses ranged from “going to a BBQ” to “buying drugs”.

Michie, director of the centre for behaviour change at University College London (UCL) and member of the government advisory body the Scientific Pandemic Influenza Group on Behavioural Science, said ministers had operated on the “back foot” for much of the crisis and needed to do more to retain the trust of the public, particularly poorer households who had no private gardens.

“We know from the evidence that if groups feel that the government is recognising the situation they are in and addressing it, they are more likely to trust and more likely to adhere,” said Michie.

Experts welcomed the call to open access to private schools’ land, with some saying that the government should consider any scheme that made it easier for people to exercise safely, rather than issuing threats. The health secretary, Matt Hancock, warned last week that the government could stop people leaving their homes for exercise if too many individuals flouted its rules on social distancing.

Professor Stephen Reicher of St Andrews University, a specialist in crowd psychology, said: “Perhaps the government could start by asking the question: ‘How do we help people to comply?’ Rather than starting from a punitive perspective, you start from a facilitative perspective.

“If you start from a presumption of goodwill towards people – how do we help? – then one obvious answer is by increasing the amount of space they can go to.”

With the link between access to outdoor space and benefits to mental health well established, Dimitrios Tsivrikos, a behavioural psychologist at UCL, said that the mental health of the poorest, typically those living in overcrowded flats with no outdoor space, was most likely to be affected by the lockdown.

In addition to requisitioning swaths of private land, Michie also urged the government to seize empty hotels and office blocks and convert them into safe spaces for victims of abuse and domestic violence.

She said: “If the government doesn’t [do more to] acknowledge that we’re not all in this together – some people are much more in it than others – and take steps to address those at the bottom of the pile, then I think that during the next weeks of lockdown there is potential for resentment, disquiet, anger, lack of adherence and even social unrest.”