Thank you so much!, LizzieL, coombs, Brodie, and amondg, for all your kind help. James Wait/e left a will; he was a successful brandy merchant, but he doesn't mention David Wait.
Coombs, I have looked at the witnesses at David's marriage to Eleanor Rice in 1763. They were James Wood and William Brodie. Brodie certainly does sound Scottish, and intriguingly, a William Brodie was baptised in Edrom, Berwickshire, just 3 miles from a family of well-established Waits in Chirnside. But at the moment, I can't find a David Wait/e that fits.
James Wood, the other witness to the marriage, has quite a common name around Middlesex. There are two that stand out. A James Wood, born in St. Martin in The Fields, is recorded in the marriage bonds as marrying a Ruth Wilcox at that church. His signature is similar to the one on David's marriage and St. Martin in the Fields, according to the author of Scotophobia, was popular with the early Scottish settlers. Thanks once again for all your kind help. I will have to keep chipping away!