The delay tends to be in the notification as this is pushed to your phone via Hik Connect. It's a free service, so that time delay (for me at least) can vary. "Most" of the time I find the notifications work great, but sometimes I've had them delayed nearly a minute.
For your security lights you would be setting your linkage action to also "trigger alarm output" and using that output to drive a relay acting as a switch for your lights. As that is handled locally by the NVR (if using the NVR alarm output) or camera (if using a camera with input/output terminals) it will be near instant with no delay so you'll have no issue there.