In political contexts, being woke means you’re aware of the problems all around you, such as racism, sexism, anti-Semitism, hate crimes, the plight of the homeless, the hungry, veterans, disabled people, the unemployed, the poor, bridges needing repair, roads needing repair, foul air, foul water, etc. Being asleep, means you willfully ignore everything that doesn’t directly hurt you.
Conservatism used to be a political philosophy. A set of ideas about governance, based on slowing change. It was the philosophy of those who were OK with how things are, and who would like them to remain that way.