“If you get Omnicron, since it is a variant of COVID, would that then provide you with natural immunity to other strains of COVID?”
The previous strains of COVID all conferred very strong immunity to re-infection by the other strains.
Omicron is so different (over 50 mutations) that the cross immunity with other strains is weaker - but it still seems pretty strong.
Out of the first 43 American cases identified and tracked, 6 had previously had COVID (14%). But that sample was tiny, and likely not random or really representative. In London, with wider spread/much higher numbers, it is running about 5% of the Omicron cases had previously had COVID.
It seems that Omicron will likely confer strong, but not perfect protection against re-infection by other strains of COVID - likely stronger than vaccination.
So, if you look at the way a “traditional” vaccine works ... a weakened or dead version of the virus is introduced to the body to create an immune response that the body can then replicate faster when reintroduced to the virus later. So Omnicrom, being a weaker version of COVID could actually be a “natural” vaccine.