I am updating some of my old Swift 2 answers to Swift 3. My answer to this question, though, is not easy to update since the question specifically asks for NSDate
NSDate
If you are looking for timestamp with 10 Digit milliseconds since 1970 for API call then, below is code:
10 Digit milliseconds since 1970
Just 1 line code for Swift 4/ Swift 5
let timeStamp = Date().timeIntervalSince1970
print(timeStamp) <-- prints current time stamp
print(timeStamp)
1587473264