A report by the Local Government and Social Care Ombudsman highlights a string of failures by the council to support the siblings, both of whom had significant needs requiring specialist support and respite placements.
The report says the children, V and W, who had previously spent time in foster care, went to live with their aunt and uncle after they obtained a Special Guardianship Order (SGO) in November 2014.
It details how their aunt and uncle did not attend court and did not see the support plan that was proposed for the children until it had been finalised.
The council's plan set out financial support for the couple "equivalent to basic boarding out rates for foster carers" but there was nothing to suggest the children would need any "therapeutic services".
However, a CAMHS report in May 2015 which post-dated the support plan, highlighted the children's diagnoses of post-traumatic stress disorder, attachment disorder, autism, attention deficit hyperactivity disorder and learning difficulties.
The couple reported that they anticipated the council would refer the children to counselling and to CAMHS before being placed with them but neither of these referrals took place.
No training for the couple was also thought necessary despite some being identified by CAMHS and social workers, the report added.
The children's uncle complained to the ombudsman in September 2016 about the lack of support provided by the council to support the children with their significant needs as well as provide respite for himself and his wife.
The report says there was no requirement for respite care set out in the support plan but the council accepted it would provide a placement for the children at a meeting in January 2016.
However, it said the council struggled to identify a suitable location for respite for W because of his significant sexualised behaviour and did not provide any respite for V.
Despite the council agreeing to a number of recommendations to support the children by February 2018, the uncle returned his complaint to the ombudsman as many of the actions had not been implemented.
The ombudsman is calling for the council to apologise to the couple by failing to provide support to their niece before she turned 18, which caused them distress.
The council was also asked to write to the family - jointly with the authority in the area where they now lived - setting out how they would be supported in future by issuing an amended SGO support plan.
It should assess the couple's training needs and provide training where appropriate and pay them any backdated respite for their niece that had not been taken together with a further £300 for the distress caused by delays, the ombudsman added.
An apology was required for the lack of explanation relating to how the amount the family would be paid was calculated with a further £100 awarded for time and trouble.
For the couple's nephew, the council was told to identify an appropriate respite placement and pay the family £250 each month until one was found as well as backdating payment for the respite provision already missed.
"The ombudsman has the power to make recommendations to improve processes for the wider public.
"In this case, the council should change its procedures to ensure it keeps complainants informed of the progress of implementing outcomes from their complaints and reviews those procedures for effectiveness," the report said.
Michael King, local government and social care ombudsman, said members of the public could only have confidence in their local authorities if they honoured their commitments.
"In this case the council's lack of urgency in carrying out its promises within a reasonable timeframe has undermined the family's trust, and calls into question its willingness to accept fault and put things right.
"This is particularly important in cases such as this where vulnerable children are involved.
"I now urge Lancashire Council to act swiftly and complete the recommendations they agreed in 2016 and those I have now made to improve both the family's situation and others who complain about its services," he said.
The county council said it accepted the recommendations outlined by the ombudsman and had drawn up an action plan to address the "shortcomings" identified in the report.
"We are very sorry for the distress our failings have caused and we have apologised fully to the person involved and their family," said a council statement.