Every time Hasina Ahmed tells her children the family may have to move, they cry. "They find it really upsetting, they don't want to leave. They are doing so well in school they want to stay where they are," she says. "Last night I didn't sleep the whole night because I was thinking if I have to move, where am I going to go? I am going to have to start from scratch.'

Ahmed (not her real name), is one of the first casualties of new rules that since April have capped housing benefit at a maximum of £400 a week for new claimants. Because she had to make a fresh claim when her abusive husband left, Ahmed's housing benefit now covers just over half of her £795 weekly rent and she now faces eviction with her five children aged between three and 14 from their rented home near Edgware Road in central London after racking up nearly £8,000 in arrears.

Ahmed's will be one of a flood of families forced to move out of privately rented homes in inner London to the outer boroughs or beyond once the government's new housing benefit rules start coming into effect for existing claimants in January.

London Councils' figures suggest there will be an exodus of 82,000 households. It says this could rise to 133,000 with the introduction of universal credit in 2013 that will cap the total benefits a family can claim.

Both changes will put immense pressure on services elsewhere, says charity the Zacchaeus 2000 Trust (Z2K). This week, it launched dedicated support for families in central London affected by the new rules and is campaigning for councils to plan strategically for the thousands of the capital's poorest families who are on the move.

"It is difficult to predict how tenants will react to not being able to pay their rent – some may stay at the risk of overcrowding, for example, or try to make up the rent shortfall themselves. But there is no doubt that there will be substantial migration and we are very worried about the strain on services," says Z2K chief executive Joanna Kennedy. "This will hit all kinds of services – schools, social services and all health services."

She adds that both tenants and councils have been in denial about what will hit them next year. London Councils says local authorities have been working hard to plan ahead, but Steve Bullock, its executive member for housing, warns: "The expected number of families who will have to move as a result of housing benefit changes and universal credit, which is more than the entire population of York or Oxford, presents boroughs with significant costly challenges at a time of acute budget restraint."

That is certainly the case in Barking and Dagenham. Because it has the lowest rents in the capital, the east London borough is expected to be the destination for many families forced to move. But Rocky Gill, the council's deputy leader, says it is already struggling to cope with a high birth rate and an influx of families who have been attracted by relatively affordable properties.

"We have serious concerns about our capacity to keep up with current demands – never mind future demands," he says. Gill says Barking and Dagenham already needs the equivalent of two new four-form entry primary schools every year. It is expanding every primary school in the borough, has converted two office buildings into schools and is looking for other empty properties to use. The worst case scenario, he warns, is split shift schooling, where some children would be taught in the morning and some in the afternoon. This hasn't been done in the borough for 150 years.

Back in central London, analysis by Westminster council suggests that up to 17% of primary schoolchildren could have to move – rising to 43% in some parts of the borough.

Angela Piddock, who recently retired as head of Westminster's Wilberforce primary school, says: "It will have taken a long time for families to build up support networks and to suddenly be cut free from them is quite devastating; the impact on children and families will be huge."

But Philippa Roe, cabinet member for strategic finance at the Conservative-run council, rejects as "scaremongering" the number who may be forced out of the borough. She says some landlords are already agreeing to lower their rents – in one case from £800 to £400 a week – as a result of the cap. She concedes, however, that some families, particularly larger ones, will have to go. "It is always difficult when people have to move but we have got into a position where people who are not working are living in properties that people who are working just couldn't afford," she says. "Something has to be done to rebalance that."

Although the impending housing benefit caps will hit London most heavily, the overall benefit cap means it is likely to fast become a national issue.

And Karen Buck, Labour MP for Westminster North, says it is not just jobless households that will be affected.

"Ministers promised that rents would fall, work incentives would improve and hardship would be avoided. In fact, rents are soaring, 115,000 more working households have been forced on to housing benefit since last summer and homelessness is rising sharply. We may still not know exactly how many people will be forced out of their homes, but we do know that no one seems prepared for the consequences of this upheaval."