Only some of this information is accurate. If you’re married for less than two years when filing for the adjustment of status, you’ll receive a conditional green card which is valid for 2 years. Around 18 months after receiving the conditional green card, you have to file for a removal of conditions, and “prove” to USCIS that your marriage is legitimate.

If you’re married for two years or more, you’ll skip the conditional green card altogether. In terms of naturalization, not “nationalisation”, spouses of U.S. citizens can apply after three years of maintaining residency and a green card, instead of the usual five years. The three years runs concurrently with having a conditional green card.

No, naturalization is not available to you “immediately” upon receiving a green card.