You are at a disadvantage when pressuring wake up.
Most fighting games have it where players who have been knocked down are at a disadvantage. For example, if you try to attack immediately after being downed, you will have whatever attack you do get snuffed in the enemy is ready. This is not the case in Demon Slayer, as the attacker is at a disadvantage. If you stay close, be patient and bait a decision instead of pressuring downed opponents.