Should traditional lifestyles become extinct, cultural heritages would be lost.
Should traditional lifestyles disappear,cultural heritages would not be passed on for posterity.
The word "extinct" is ok , and other words like "disappear" or "die out" sounds better for above.
Example : Many languages are getting extinct/disappearing/dying out.
Living styles is not good . It should be lifestyles, which are styles of living.
You usage of "neither" is wrong : it should be "not".
Should traditional lifestyles die out , cultural heritages would neither be preserved nor remembered.