***** NOT A TEACHER *****
May I share a few thoughts?
1. You used "went." So I think that we need to use "was," not "is," in both cleft sentences.
2. "My GP was who I went to see." Personally, I would not use such a sentence, but I found a similar sentence
in Mr. Michael Swan's popular Practical English Usage:
"Mary is the person who keeps a pig in the garden shed." Come to think of it, your sentence would sound better to me
if you said, "My GP was the person who I went to see." (Super strict teachers would require "whom," but there are
very few strict teachers anymore, so we won't worry about it.) Anyway, probably most speakers would use "that."
I think that the term "cleft sentence" usually refers to the kind of cleft sentence that consists of IT + THAT. Mr. Swan
tells us that we may use "who" for a person if we want to emphasize the person. But usually, a so-called cleft
sentence would use "that." It is almost a formula:
"It was my GP that I went to see."
Let's use this sentence: The Parser loves to eat ice cream every day.
It is The Parser that/who loves to eat ice cream every day.
It is ice cream that The Parser loves to eat every day.
It is every day that The Parser loves to eat ice cream.
The Parser is the person who/that loves to eat ice cream every day.
What The Parser loves to eat every day is ice cream.
Every day is (the day) when The Parser loves to eat ice cream.
- For Teachers