Train from Paris to Brussels

The UK is not in the Schengen zone, which is why there is the passport check/screening (you enter Schengen when you enter France). I don't think there is any screening in Paris at the station (unless they've added some in recent years due to the attacks there). The UK train screenings have always been far more vigorous in my experience. Last year in Italy for example I simply hopped on the train in Genoa and traveled on to Nice - no screening at all.

In any case, I would still expect minimal security checks if any at all.

From Thalys FAQ: At Paris Nord, security gates are in operation at the entrance to Thalys platforms. Before boarding the train, each passenger must pass through these security gates and have their luggage scanned.

They've suggested getting to the station at least 20 minutes early. (In Belgian stations, security checks are more random.) Gare du Nord is large, but also a relatively simple layout. Be sure to have luggage tags with your name and address.

Andrew is correct, there is no passport check as both France an Belgium are in the Schengen Area.
There is a bag scan. This is done at the entrance to the platform and is quicker than the scrum on the Eurostar. There is no minimum time specified, just be at the platform 15-20 minutes before departure as normal.

I expect that you are on your way, but if not I'd leave a little earlier because today is May Day, and things tend to slow down on May Day. If I were going I would be there half an hour ahead, at least. Given the general industrial relations climate in Paris at the moment things may slow even more than on a "normal" May Day.

Thanks all for your advice. We didn’t have any problem at all. We arrived an hour early and the security check took only about 10 minutes. It’s helpful how they do it at the platform for each train. They also did this at the airport in Amsterdam and was much better than the system used in the US which is very long and stressful.