Would you have been covered on your old policy? And are you also covered on your new policy? If so you get the minimum cover out if those two policies while you serve your waiting period.
So if your Laya policy covered a procedure with a €500 excess and Aviva covers same procedure then you'll only get the Laya coverage for that procedure for two years (5 years for older customers).
If your Aviva policy doesn't cover the procedure, but Laya did, then you're not covered so get nothing.
If Aviva policy covers something but Laya didn't, then that's an increase in benefits and so you will not be covered for 2/5 years depending on age.
When you join up with Aviva you'll be ask who your previous insurer was and what level of cover you had. If you make a claim this will be verified with Laya and the claim paid appropriately.
You cannot get around this by switching insurers - that would defeat the whole point of the open market and all legislation to allow you to choose equally between insurers without penalty. Remember waiting periods are for insurers benefits.
Saying that, those are the max waiting periods and there's nothing to stop any insurer waiving them completely if they want.