Andy,
This question has been asked in the past by the club itself which believed that it was originally designed by Tilly but finished by Robert White. Unfortunately, Tilly never worked on it.
The confusion arises due to its name, the Water Gap Country Club, which is similar to one that Tilly did design (It's name escapes me at the moment & I'll have to look it up & post it later).
Water Gap was originally named the Wolf Hollow Country Club. I believe it was late in the '30's when it closed and then was re-opened after a new owner bought it after the war, but I'll have to look that up as well. It opened for play in 1924 and was definitely designed and constructed under the direction of Robert White. White advertised it as such in Golf Illustrated in 1925.
It was at Wolf Hollow that the Eastern Open was held for a number of years and where Walter Hagen won it in 1926. In the August issue of Golf Illustrated in 1926 there is a photograph of the 2nd green which clearly shows how dramatic the loss of putting surfaces from then to now is. It also shows how undulating the outer third is and what a difference it would make.
There is a small piece of the article in which it appears that talk of the course and the greens themselves. "At the Delaware Water Gap, Pennsylvania, is the eighteen-hole course of the Wolf Hollow Country Club. It is now in its second year of play, and the greens and fairways are clothed with such good turf that the methods used to produce it and to maintain it are of interest. The soil is of average quality - a medium clay loam full of stones of all sizes from ton boulders down; stone collection and disposal were, therefor, a big problem. The greens were shaped out of the sub-soil as usual, but from then on the procedure was not so orthodox. In the first place, manure was disced into the rough's grading and fertilizer also was worked into it; then nearly a foot of top-soil was spread all over and this too was manured and fertilized. Now the laborers were marched up and down and across the greens until every inch had been well-trodden; this to compact the soil and to induce soil moisture to work up from below."
The reason this club gets confused with being a tilly is Robert White himself. He worked for Tilly and learned much from him. He had just left him a few years earlier and so it was assumed that Tilly had done the original design and was planning on White to build it, but it is a solo and complete White design, and one which he would be proud of. The course itself has many aspects that look like Tilly's handiwork, but that should be expected.
It would be wonderful if the club did some serious research and restored the greens to their original dimensions...