Is it correct to say (or write) "the only one" with a following noun, e.g.
It is the only one car that has such speed.
I promise it's the only one soldier who can do this.
Are those sentences correct?
No. "The only one" refers to the same thing that e.g. "the only soldier" does, and lumping two words with the same meaning together like that does not work well at all. Instead, for this sort of emphasis, you could use "the one and only [X]", such as "I promise it's the one and only soldier who can do this." But in these contexts that level of emphasis doesn't seem necessary; the simple factual nature of the assertions is quite strong enough.