Yes! These are separate processes. You are time barred if you have not notified a compensation event within 8 weeks (if it is one you are responsible to notify). The time bar commences from when you knew it was a compensation event - not when you first raised an early warning.
You might notify an early warning about a potential problem. 12 weeks later it becomes a CE. It is from that point that you have 8 weeks to notify. The fact you raised an early warning is irrelevant at this point in terms of the time bar.
A point of discussion is where an early warning has been raised and the event becomes a compensation event. Where it is obvious that the PM was aware of the event, would the ‘time bar’ issue still apply to the Contractor’s notification? Although it implies the PM is obliged to notify matters which he (essentially) instructs, does that mean that the PM can adopt an approach of ‘sitting back and watching from the sidelines’?