Contractors carrying out urgent checks to 17 schools in Edinburgh closed suddenly over substantial safety fears have admitted they still have no clear timescale for their full reopening.

Nearly half of the 7,600 children left stranded in Edinburgh after the emergency closure of the secondary, primary and additional needs schools at the weekend will be back in class from Wednesday, after an urgent search for alternative accommodation, the city council said.

City of Edinburgh council said they included the 2,000 senior secondary school pupils starting their exams later this month. They will either return to their own schools or be rehoused at other nearby high schools on Wednesday and Thursday this week.

The remaining 4,300 children still to be rehoused would be given alternative places by Tuesday next week, following a one-day Edinburgh-wide holiday on Monday.

Pupils at Oxgangs primary, where the collapse of a substantial brick wall during a gale in January sparked off the crisis, will return to classrooms at another site on Wednesday, as will children at St Peter’s primary, where similar defects were found, and two other primaries.

The defects, which involve missing “header ties” that bind brick walls to stronger structural walls, have also been found in two other secondaries, Craigmount and Gracemount, with urgent surveys underway to check the other 13 buildings.

But the main contractor responsible for the management of all the privately financed schools, Amey, said they were still unable to offer a timetable, costings or a schedule for the remedial works involved due to the scale and complexity of the survey task.

“We have completed the first phase of initial remedial works in three schools over the Easter break, and we will continue to support the council with the on-going investigation and additional works requested across the Edinburgh schools estate to ensure they are safe for use and can be reopened as quickly as possible,” the company said.

Andrew Kerr, the council’s chief executive, said finding alternative places had been “a huge logistical exercise”. He added: “I fully recognise the significant inconvenience to parents caused by these closures and I want to thank them for their patience as we continue to work through this issue.”

The crisis has sparked fresh demands from unions and party leaders, including first minister Nicola Sturgeon, for reviews into the safety of other privately financed schools and public buildings, and a rethink about the heavy reliance on private financing for public sector projects.

The Guardian has established that the private consortium which built, maintains and owns the schools, Edinburgh Schools Partnership, is itself owned by four investment and private finance companies via a nominees company run by Bank of Scotland.

ESP was originally set up by Amey, the building firm Miller Construction and BoS in 2001 to run the £360m contract under the then Labour government’s public-private partnership (PPP) programme to build or refurbish the 17 schools and a community centre.

Highly profitable – it made £3.4m in after-tax profits in 2014 against a turnover of £10.4m and £1.9m profits last year – ESP’s largest shareholder is the London-based finance house Semperian PPP Investment Partners, with 33% of its shares.

The second largest shareholding, 30%, is held by Edinburgh-based Aberdeen Asset Management, whose founder, Martin Gilbert, is a close friend and supporter of former Scottish National party leader Alex Salmond. Gilbert endorsed Salmond’s bid to be first minister in the 2011 Holyrood election.

A collapsed wall at Oxgangs after winds of more than 90mph struck in January. Photograph: Andrew Milligan/PA

A third investment fund, Palio (No 19) Ltd managed by John Laing Infrastructure Fund, holds 20% and PFI Infrastructure Finance Ltd, managed by 3i, owns 17%. All four funds shared in a dividend pay out of £1m in 2015, and £800,000 in 2014.

ESP’s company accounts show that Edinburgh council still owes it £72m – a figure which excludes the annual management and maintenance fees for the 17 schools. The council has paid it £10.5m in interest alone on construction debts and long term borrowing over the last two years – nearly half of ESP’s overall income.

ESP’s shares ownership structure is concealed by a nominees company run by BoS, an original investor and partner in the company.

Mary Alexander, deputy Scottish secretary for the public sector union Unite, said these profits underlined Unite’s long-standing worries about the long term costs of the widespread and continuing use of private finance schemes for public buildings.

“These figures reinforce our view that PFI/PPP might deliver healthy profits for contractors but it is detrimental to health and safety, and painful for the public purse when services are being slashed on the back of a never-ending debt burden,” she said.