This is my understanding:
Ford Contracted Eric Broadley as a consultant since Eric Broadley's shops had built a rear engine V8 powered sports car called the Lola. Ford used 2 of his Lolas as only the 1st TEST MULES for a rear engine Ford Powered car. Broadley spec'd a few components for the new Ford GT (GT40) that were the same as those used on the LOLA, the Boranni Wheels and Collotti transaxle. Both failed on the GT. The Ford GT40 body was designed by Ford's Roy Lunn, with Ford's Eugence Bordinat making the first clay models. Initial wind tunnel testing was done at the University of Maryland. Ford's Klauss Arnig designed the suspension using a computer, a FIRST in automotive design. The first GT40s were built by FAV in England. The first GT40s were complete racing failures as they were a DNF in every race they entered. Fast forward a couple of years to the LeMans winning MKII. A complete redesign and construction was all done by Kar Kraft in Michigan.
Although the aluminum was bent in England, most of the modifications were by Ford Dearborn and Kar Kraft. Shelby did not want to get involved as he wanted his Cobra Daytona Coupe to run with his name on it. Ford just contracted him to run the team after the work with Holman and Moody. Bruce McLaren had been involved as a development driver from early in Broadley's team. Bruce McLaren then went on to develop the Aluminium Roadster in Britain.
Lola didn't do development work although Ford did hire Broadley for a short time.
The cars were originally built and developed at a special factory, Ford Advanced Vehicles, that they set up for the purpose with a team gathered from several sources to develop and build the cars. Broadley's involvement was fairly short lived as he had some disagreements with where he thought the design should be headed and so left the team. Cars were built at FAV in the UK and shipped to Shelby for final tuning and track development work. The first fully USA designed and built car was the MKIV. The Lola was certainly the inspiration behind the GT40 and many design features from the Lola can be found in the Ford but other than that, Lola as a company did not have anything to do with the GT40.