The report, Child Protection in Faith-Based Environments, calls for the government to establish a national registration scheme for such schools, also called madrasas, to ensure that they follow child protection measures.
The madrasas teach basic Islam to about 100,000 Muslim children in Britain in after-school classes, with some of the mosque schools having attendances of more than 500.
But only a handful of local authorities have insisted the schools meet their child protection obligations under the Children Act 1989.
The leader of the Muslim Parliament of Great Britain, Ghayasuddin Siddiqui, said: "Sweeping the issue of child abuse in UK madrasas under the carpet is not a solution.
"If nothing is done now, we may face an avalanche of child sex-abuse scandals, decades afterwards, similar to those that rocked the Roman Catholic church in the 1990s."
Siddiqui also wrote in the report's introduction: "The Muslim community is at present in a state of denial - denial of the fact that child abuse takes place in places of worship including in mosques, madrasas and families.
"If madrasas are left on their own, it is likely that due to poor understanding of child protection law and practice, a large number of Muslim children will remain exposed to significant risk of harm."
He said anecdotal evidence suggested that around two-fifths of the teachers in madrasas hit or scolded their pupils, and that reports of between 15 and 20 cases of sex abuse a year understated the true level.
"It would have been surprising if there were no cases of child abuse as Muslim societies are, after all, like any other human society"
leader, Muslim Parliament
of Great Britain
"It would have been surprising if there were no cases of child abuse as Muslim societies are, after all, like any other human society," Siddiqui said.
A Labour member of parliament, Anne Cryer, will speak at a conference in London on Wednesday to promote the report.
She said: "Madrasas are no different to any other organisation that works with children - Criminal Records Bureau checks and child protection procedures must be in place."
She said she had had reports of physical abuse in madrasas in her West Yorkshire constituency, Keighley.
"Failing to protect the children in madrasas because of cultural sensitivities is nonsense. Are we saying that British Asian children are not entitled to the protection of the law?" she said.
The education department said the government's child protection laws would require employers to check "all those who frequently teach, care for, or supervise children, including those in madrasas".
Muslim body calls for national register of UK Madrasas to counter child abuse - press release, 22 March, Muslim Parliament of Great Britain