Mark a very interesting question.
Braid was known for his bunkering, often called in to add some to and existing course. Fowler’s bunkers at Walton Heath were famous and the members called them coffins. I would struggle to see any stylistic similarities however.
Braid was known for dogleg holes. Not really a big feature at Walton Heath.
A couple of years ago I reported on here that there was a revision to Moreton’s book on Braid coming out. I believe the co author told me they’d discovered Braid’s first courses date back to the 1890’s, well before Fowler started.
Braid was a quick study and he left plans behind. Fowler rode the Heath literally for years before work commenced.
Also if you look at the set up of Walton Heath its quite clear that Braid had his own sphere of influence but that Fowler was in command. He was Managing Director and as Tom points out Braid was never officially allowed to influence the design. His workshop was in a separate building to the clubhouse, into which he could only enter if invited. The two came from the most distant strata’s of society. Braid was a servant of the club, albeit a handsomely rewarded one. Fowler was happy to be the Master.
I feel any influence must have been subconscious.
I can just imagine that there may have been a few clubs who approached both parties who then became rivals for the commission. It would have caused friction.
Mark is there a reason for your question?