Smith worked with Watkins at both Aston Villa and Brentford.
After his England winner in the semifinal against Holland, Smith told BBC Sport: "The reaction was to dance around my apartment in Charlotte to be honest.
"It was a fantastic finish, a great ball from Cole Palmer, but something that I have seen Ollie Watkins do an awful lot.
"It wasn’t instinct, that’s repetitive practice. One that Richard O’Kelly will be happy with because I’ve seen him taking Ollie Watkins a lot on finishing sessions and scoring lots of goals like that.
"I actually saw him play when I was managing Walsall and I went to watch him play for Weston-super-Mare. He was on loan there from Exeter. A friend of mine who was at Exeter at the time had told me about him.
"I liked his attitude, his application and he had all the tools. I think the biggest problem was he had a little bit of imposter syndrome at the time and wasn’t sure where he was. He has just grown and grown as a player.
"There are not many players that I buy twice and I bought him at Brentford from Exeter and then for Aston Villa. He became the record signing for Villa and he never let me down.
"He has never let me down and he’s just grown. He made his England debut while he was with Villa, while I was there with him. It’s just been a pleasure to see, you know, how he’s grown."