I finally decided to give HP Support a call to see if they had a solution. After an ungodly wait on an antiquated support call system with reminders every minute that all their agents were busy and with no indication of an expected wait time I managed to get through to them.
They recommended unplugging one of the two monitors to see if the problem would still occur. It didn't. Swapping it and trying the other monitor as the primary also didn't show the blinking. Connecting them both back up to the macbook pro also then didn't show the problem !
Seems like the issue only happens when the monitor activates sleep mode. I didn't realize that I had to wait for not only the screen to blank out but the monitor to say it is going to sleep. The issue still occurs. The HP monitor has a power control option and an auto-sleep mode. Setting the auto-sleep mode to off prevents the problem from occurring. I likely didn't have this issue with the old monitors as they didn't have a power saving sleep mode. Unfortunately there is no auto-sleep mode delay. My typical use case is locking the computer and going to grab a coffee which is typically a one minute away period but is quick enough for the monitors to auto sleep. The blinking is annoying enough that I'll just keep the monitors setup with auto sleep set to off. Ideally I'd be able to power save and have them auto sleep unfortunately it is just too annoying.
Do other people have auto sleeping monitors that when waking take 3 blinks and around 10 seconds for the monitors to settle back down?
Be interested to know if 2 external monitors on a macbook pro that have auto sleep will always have this issue, if the issue is HDMI related, or HP monitor limited. I'll play around with this some more and see if having it only using a single monitor changes things, maybe try out the old vga cable (these monitors are at 1920x1080 so maybe I wouldn't see any difference not sure on the analog vs digital signal difference with the different cables).