I have appreciated what others posted so here's how it went for me Sunday morning (American adults headed to USA).
One thing I had had not known - TAP does not make you wait until 3 hours before your scheduled departure to check your luggage. That limit may apply early in the morning when nobody is working yet, but there were no limits around 8am.
We dropped our bags pretty quickly, went thru the xray security, and then walked thru duty free and down the length of the airport to get in the incredibly long passport control line. There was a friendly airport employee in a fluorescent vest there. I asked him how long it would be and he laughed and said well, at a certain point they will just open the borders, so it is impossible to estimate. Indeed, after about 30 minutes, the line sped up so much that we were walking briskly and when we got to the electric gates, which were all blocked open, guards were shouting “walk thru”! We didn’t scan or fingerprint. At each “e-gate” a person stood stamping passports the old fashioned way but not stopping to talk to anyone.
But this was at a time of day with multiple big flights departing. At a different time it might be different.
Also fwiw if you pay for “fast track” xray security, it apparently is one floor below regular security. We paid but didn't find it. In any event the normal version wasn’t too bad.
All in all we had plenty of time after passport control. It’s just a shame we couldn’t spend time in the main section of the airport with more lounges and restaurants. The food options in the post-passport-control section had 25-minute lines just to buy pre-made sandwiches and there was nowhere to sit. People lined the walls. There was a long line for the TAP lounge, the only one that is after passport control.