The reports published in London on Thursday by the parliamentary intelligence and security committee amount to one of the most damning indictments of UK intelligence, the Guardian newspaper reported.
The committee report revealed links to torture and rendition were much more widespread than previously reported.
British intelligence agencies continued to supply intelligence to allies despite knowing or suspecting abuse in more than 200 cases, the report found.
The chairman of the committee, the MP Dominic Grieve, said British agencies knew of incidents that were plainly unlawful, with the report concluding that intelligence agencies were aware at an early point of mistreatment of detainees by the US and others.
The committee concluded that a blind eye was turned to routine mistreatment by US authorities, although British intelligence agencies became aware from an early stage that this was going on.
The new report has already led to calls for an independent, judge-led inquiry.
May said, in a written response to the House of Commons, that it was important to begin by noting the context in which the British government, including the security and intelligence agencies and armed forces, were working in the immediate aftermath of the 11 September 2001 attacks, and the deployments of Armed Forces personnel to Iraq and Afghanistan.
The report said that while there was no evidence of British officers directly carrying out physical mistreatment of detainees, the intelligence services MI6 and MI5 were involved in hundreds of torture cases and scores of rendition cases.
Rendition involved sending a person from one country to another for imprisonment and interrogation, sometimes methods such as torture that would be illegal in the country doing the rendering.
May added: "The UK responded, alongside its international partners, to the tragic events of 9/11 with the aim of doing everything possible to prevent further loss of innocent life, both here and overseas. In Iraq and Afghanistan, the priority again was preventing loss of life."
She added: "With the benefit of hindsight, it is clear that UK personnel were working within a new and challenging operating environment for which, in some cases, they were not prepared."
May said it had taken too long to recognize that guidance and training for staff was inadequate to understand fully and take appropriate action on the risks arising from British engagement with international partners on detainee issues.
"The agencies responded to what they thought were isolated allegations and incidents of mistreatment, but the ISC concludes that they should have realized the extent to which others were using unacceptable practices as part of a systematic program. The agencies acknowledge that they did not fully understand this quickly enough and they regret not doing so," added May.
May said the British government does not participate in, solicit, encourage or condone the use of torture or cruel, inhuman or degrading treatment for any purpose.
She added that detainee-related work remains important and at times difficult, but intelligence and armed forces personnel are now much better placed to meet that challenge.
May has called for proposals to be made to the government about how the guidance to personnel could be improved, taking account of the views of the committee as well as those of civil society.