they're not working on the other parts as far as we know
I agree, but we wouldn't know. Iran is a modern world power that's perfectly capable of doing things in secret that don't fit in a garage.
We know with certainty that Iran has enriched a stockpile of 60% uranium. That's not sufficient to say that nuclear capabilities are imminent, but it's enough to say that they have long term goals in that direction. (Which makes perfect sense, and should not itself be provocative).
You also need someone who understands the engineering well enough to actually do it, particularly without testing
The US did it 75 years ago with no instruction book. Iran's population is about the same as the US's was at the time, and they have the benefit of all those years of manufacturing advancements. It's borderline racist to assume this would be a problem for them.
They don't have icbms
They have IRBMs, and Israel has been far from 100% successful at shooting them down. That's with heavier payloads that have to reach the ground before detonating. Their IRBMs also include a small number of more modern systems that are nearly impossible for Israel to shoot down.
they could just lie and say they have it
Like in high school? "I swear, she used her tongue and everything!". I'm not so sure that's going to buy them much leverage. (Which would be the real point of having them.)
Do Russia and China have working nukes? What about India and Pakistan? Do the US nukes even still work? No one is sure.
I don't think there is any plausible doubt about any of those but Russia. Even Russia is still certain to have some functional nukes. The only doubts are about how many. Also, Fission bombs have a really long shelf life. It's fusion bombs that require tons of ongoing and expensive maintenance.