So, upgrading to 10.11.1 did not fix my issue with a lagging wireless Apple keyboard. Upon first wake from sleep the keyboard exhibited the same symptoms.
I've been searching this and other forums for insights since my problem started. I found an interesting thread on these forums that may have helped. The poster was having troubles with a Magic Mouse stuttering and losing connection. He finally solved his issue by moving (not copying) the plist files for Bluetooth onto his desktop and restarting his Mac. The plists were recreated upon restart by OS X and his problem was solved. Seems the Bluetooth connections profile had been corrupted and the fresh ones created by OS X restored the use of his mouse. I gave it a try and so far my keyboard has been working normally when coming out of sleep. My Apple keyboard, mouse and trackpad were automatically reconnected at startup, but other Bluetooth devices have to be re-paired.
If you want to try it for yourself here are the steps the other poster outlined:
"Move the following files to the Desktop and restart:
- /Library/Preferences/com.apple.Bluetooth.plist*
- ~/Library/Preferences/com.apple.Bluetooth.plist*
- ~/Library/Preferences/ByHost/com.apple.Bluetooth.*
It’s important to move (not copy) the files. This forces OS X to re-create the files on reboot. (I could have just deleted the files but I wanted to keep them around as backups in case something went wrong.) Since doing this, my Bluetooth devices have been happily connecting and disconnecting appropriately and I have no more stuck devices in my Preferences"
Thanks to Matthew Bass for the above suggestion. I have placed my MacBook Pro into sleep mode several times and the keyboard continues to work as it should. Previously the keyboard lag would always be present when I woke the computer from an overnight sleep. This morning my keyboard was working normally. I am hopeful that the issue has been resolved, but will report back to let you all know how it goes over the next couple of days. I was just about to take the plunge and do a clean install of El Capitan (which probably would have solved the issue also) but hopefully won't need to go through the hassle now!