Mr McLoughlin said "personal meetings" with those families affected – attended by officers and support staff from Hull City Council and East Riding of Yorkshire Council – began last week and would "take time to complete".
“Some families may want to receive the ashes, and others may not," he added.
"There is no right or wrong answer and the impact... will differ for each individual family, but it should not be underestimated and we will do all we can to support those throughout what is an incredibly difficult time.
“I want to thank the families and everyone in our communities for their understanding as we continue this complex investigation, and we will endeavour to provide answers and updates as and when we are able to.”
'Very angry'
Richard Shaw, from Woodmansey, near Hull, used Legacy Independent Funeral Directors for his wife Rita's cremation in October 2023. In December, he was given an urn containing what he was told were her ashes.
In March, he learned that Mrs Shaw's ashes had been found in a mortuary.
Mr Shaw said of the latest development: "I have to believe these latest ashes are Rita's or I'll go crackers.
"I am very angry."
A 46-year-old man and a 23-year-old woman, who were arrested on suspicion of prevention of a lawful and decent burial, fraud by false representation and fraud by abuse of position, remain on bail as police inquiries continue.
A 24-hour Victim Support helpline - 0808 281 1136 - has been set up for anyone affected by the investigation.
