I would also say: Take the Eurostar from London St. Pancras to Gare du Nord (takes about 2 1/2 hours) and continue by Thalys from Gare du Nord to Cologne Main Station / Köln Hbf. (takes about 3 1/2 hours) - both nonstop. If I am not staying right next to LHR or STN / CDG or ORY, I would not think twice about that.

Another more scenic (but longer) routing you might consider, depending where you are staying in Paris: From Paris Est to Karlsruhe (for example TGV InOui leaving 10:55), change in Karlsruhe to an IC/EC train (for example EC 8 leaving 14:12) going along the river Rhein and the UNESCO world heritage "MIttelrheintal". For the second part, you might read seat61: https://www.seat61.com/places-of-int...-the-rhine.htm