Trains from Lille to Rouen without going to Paris

The only trains that go from Lille to Rouen without going to Paris are TER regional trains which do not take reservations. Think commuter trains. But you will get on the train regardless even if you have to stand. That would be unlikely, since the train originates at Lille Flanders station and you will arrive 2 hours before train time, so you would be the first people to board the train. Speaking of that, you will have to go from Lille Europe station, where the Eurostar stops, to Lille Flanders station, either by walking, bus or taxi. Also you only have 6 minutes to make the change at Amiens, though that train also originates in Amiens.
You could also wait one more hour and get a direct train from Lille Flanders to Rouen.

Thanks, Sam. You have greatly eased my worries.

We have already purchased France Rail Passes.

For the TER train(s) you have listed you just get on, they don't take reservations. Still, point-to-point fares are often much less than the cost of a railpass. An adult ticket for the journey you listed is around $40. If you are taking any trains that require reservations keep in mind what RS says

Be aware that most trains with compulsory reservations limit the number of seats available to passholders (most notoriously France's TGV trains, which also don't let passholders book seat reservations less than three days before departure). Along some of the most popular routes, such as between Paris and Italy, direct trains run only a few times per day; these can sell out weeks ahead (and the overnight Paris–Italy trains don't accept rail passes at all).
https://www.ricksteves.com/travel-tips/transportation/trains/reservations

I was able to use my France Pass and book a reservation on the TGV, so all is good. The local TER arrived a bit too late to suit us. I am trying to make it as easy as possible for Mom. Thanks for the info.