Ahmad Mulakhil has been charged with two counts of rape, while Mohammad Kabir has been accused of kidnap, strangulation and aiding and abetting the rape of a girl aged under 13.
Mr Mulakhil, 23, appeared before magistrates in Coventry on 28 July, while Mr Kabir, also 23, appeared in court on Saturday.
Both were remanded in custody.
Leader of Warwickshire County Council George Finch told the same news conference on Monday morning that he was “begging” for information to be released following the charges.
Finch, who at 19 became the youngest council leader in the UK and represents Reform UK, said he had contacted the chief executive of the council, Monica Fogarty, saying he wished to speak to Warwickshire Police “to urge” them to release the immigration status of the first man charged.
“I was begging for this to be released, screaming, phoning, asking [for the information] to be released”, he said.
Following the charging of the second suspect, Finch said he wrote a letter addressed to Home Secretary Yvette Cooper and Warwickshire Police’s Chief Constable Alex Franklin-Smith calling for the immediate release of the two suspects’ details.
Finch published the letter on his social media accounts on Sunday, in which he said Ms Fogarty had told him that Mr Kabir was an asylum seeker living in a house of multiple occupancy (HMO).
Speaking on Monday, Finch said he would be working to “fight against” houses of multiple occupancy that “have been put up to house illegal immigrants”.
He also claimed Reform UK needed to “change things” and are “the last line of defence against the blob, the cover-ups”.
When asked if police should release the ethnicity of people charged with offences, the prime minister’s official spokesman said the police and courts were operationally independent but the principle was to be “as transparent as possible”.
“We’ve always said and continue to say that transparency is important,” he said.
“That is our position. For police up to central government, we should always be as transparent as possible when it comes to cases.”
He added: “This is clearly a deeply upsetting and distressing case which the public are right to feel shocked and angry about.
“In relation to this case, the individuals have been charged and we are now in a live investigation.”
In a statement, Warwickshire Police and Crime Commissioner Philip Seccombe said: “It is essential to state that policing decisions – such as whether to release details about a suspect – must follow national guidance and legal requirements.”
He added that he would not speculate on the personal circumstances of those involved while court proceedings were active.
Chris Watkins, Labour leader of Nuneaton and Bedworth Borough Council, said it was a criminal investigation and it would be inappropriate for the authority to comment further.
“Our deepest thoughts are with the victim and their family,” he added.
“We recognise that the community is upset, but this incident is not reflective of the borough.”
Jodie Gosling, Labour MP for Nuneaton, said she was “horrified” by what had happened and her thoughts were with those who had been affected.