In AmE we prefer the object form in short sentences like this.
A: Who did this?
B: I did/It was me.
Or beginning a phone conversation, "Hi, it's me. What are you doing?"
I think when we hear 'it is I' and 'it was he', it sounds like reading out of an old book, or like affected speech.
An alternative to that form as our sentences get longer, is to change "It was me who had given her the watch." into "I'm the one who gave her the watch."
It sounds more usual to begin sentences in this way.
"I'm the one who..." instead of "it is/was me who..."
"He's the one who..." instead of "it is/was him who..."
"They're the ones who..." instead of "it is/was them who..."
(not a teacher, just a language lover)