Has anyone taken a train from Lucerne to Lille?

Yes (but not all in one day, because I live in Basel which is between Luzern and Lille).

You can look up train times on the SBB website: https://www.sbb.ch/en

The quickest route would be Luzern - Basel - Paris Gare de Lyon - métro - Paris Gare du Nord - Lille. This is roughly one train every 2 hours. This is the route the SBB website will give you.

But it is also possible to go via Strasbourg, which would bypass Paris and avoid the métro. But I think there is only 1 train a day on this route:

Luzern, dep 11:54, IC 21 672, Basel SBB arr 12:55
Basel SBB, dep 13:21, TE2 96282, Strasbourg Ville arr 14:39
Strasbourg Ville, dep 15:08, TGV 9894, Lille-Europe arr 17:58

Thanks for your reply. We would really prefer going via Strasbourg to Lille, we do not want to travel through La Gare. If Strasbourg is not possible, then North of Paris? I see that SBB will not accept reservations for that route from Lucern to Lille. We have a certain day to travel, and it says it does not go on that day. Is there another place I can search? Many thanks for your help.

Try https://www.trainline.eu/
They sell French tickets, but not Swiss tickets. So buy Basel to Lille on Trainline. And either buy Luzern to Basel at Luzern station, or on the SBB website. Swiss tickets (unlike French) are not specific to a train, they are valid on any train that day and there are no reservations, so there is no disadvantage with buying on the day.

What is your travel date? SBB (Swiss site) may not have all legs for sale equally early and probably does not sell the leg from Strasbourg to Lille, since that's completely within France (SNCF). You can also use the German (DB) link at Looking Up Train Schedules and Routes Online for schedules only, not to book tickets in these countries.

Even Google Maps finds a 6 hour trip via Strasbourg, on some days. Although most ticket websites know this, I wonder if there's a search problem with the fact that there are TWO big rail stations in Lille? (Adjacent and connected.) Look at your drop-down box contents very carefully before you select your destination.