On further reflection, the most obvious candidate for providing the private campaign funds would be CSM, the sports marketing firm:http://www.csm.com/about-csm/From the Guardian:
Coe has been involved with Nike since 1978 but only took up his current role as an international ambassador, focused anti-obesity campaign Designed to Move, in 2012 after the London Olympics.
The double Olympic gold medallist has been adamant that he won’t be pushed into giving up the role, insisting it does not represent a conflict of interest. But he has said he will review all his arrangements within his first 100 days.
The role is believed to pay around £100,000 a year but is also thought to be linked to a complex multimillion-pound earn out deal that Coe signed when CSM, the Chime sports marketing arm he chairs, was sold to Sir Martin Sorrell’s WPP and private equity firm Providence with the rest of the company this year in a deal worth £374m.
http://www.theguardian.com/sport/2015/nov/24/sebastian-coe-role-eugene-2021-bidding-processAlso:
http://www.theguardian.com/sport/2012/oct/26/lord-coe-windfallBeing president of the IAAF is an unpaid position, although Diack apparently found a way to make it a paid (under the table) position. If UK Sport is buying influence by funding Coe's presidential campaign, whoever else funded the campaign is presumably also trying to buy influence. If Coe is an employee/owner of CSM, presumably CSM thinks it is profitable to them to have Coe as president of the IAAF. And the connection between CSM and Nike needs to be spelled out.
It's not at all clear what is going on behind the scenes but whatever it is needs to be public so that the track world can judge whether it is appropriate and actually benefiting the sport and its athletes.