Sony has clarified on exactly how subscription upgrades will work with the new PlayStation Plus 3-tier plans.

Sony plans to merge PS Plus and PS Now on June 13. So what if you're already subscribed to both services? Sony confirms that these users will get access to the higher-end PlayStation Plus Premium tier.
Existing subscribers of both PlayStation Plus and PlayStation Now will migrate to PS Plus Premium at no cost, unlocking access to 740 games across six console generations including PS1, PS2, PS3, PS4, PS5 and PlayStation Portable.

Sony had previously said that PS Now subscribers would be upgraded to PS Plus Premium: "PlayStation Now customers will migrate over to PlayStation Plus Premium with no increase to their current subscription fees at launch."
It's unclear whether or not current PS Now subscribers will still get upgraded to PS Plus Premium.
Remember, do not subscribe to the PlayStation Plus Extra tier. Our analysis shows it is not worth subscribing to.
Here's what Sony's new FAQ says:
I'm a PlayStation Plus member. What will happen to my subscription when the new PlayStation Plus launches?
When the new service launches, your membership will roll into PlayStation Plus Essential and you will retain access to the same PlayStation Plus benefits you enjoy today. Your recurring membership fee and payment date will remain the same as well.
If I am subscribed to both PlayStation Plus and PlayStation Now, what happens to my subscriptions and payment date?
If you are subscribed to both services when the new PlayStation Plus launches, you will be migrated into the PlayStation Plus Premium membership plan and you will have a new single payment date based on the longer of your two subscriptions. You will be notified via email of your new payment date and subscription fee.