Gare Nord to Gare de Lyon with wheelchair

Not all taxi's in Paris are small cars. Most are mid size sedans, and mini-vans are often at the train stations in line. You should have no problems.
I know where you're coming from, as my wife was in a wheelchair a year ago. I towed her 21" rolling bag behind the wheelchair, and I had a 21" backpack suitcase on my back. We went to France, London and all over Ireland without the assistance of any porter or help. There again, I've got the strength to do it. She's since got a replacement knee, and is able to hike with the best of us in cities. I'd go with 2 carryon bags and retire any large suitcase.

Yes, there are separate gates for entry with things like wheel chairs and strollers. The easiest way would be to take the escalators down to the southbound RER D platform and take the train two stops to Gare de Lyon. These are all roll-on-roll-off stations. Then up the escalator or elevator to the main train platforms.

Paris by Train has good information to help with this if you choose RER. It includes photos of the insides of the stations to help you preview what you'll be looking for.

But a taxi would be easier, and you shouldn't have trouble finding one big enough for two people, a folding wheelchair, and luggage. The taxis are on the right after you leave the train area -- go out to the curb where they're lined up, ignore anyone inside the station who offers you a ride.

Yes, there are separate gates for entry with things like wheel chairs and strollers.

I feel like this isn't the case at every single metro entrance — nor at every exit.

Ah, but Gare du Nord and Gare de Lyon? I think so, since RATP shows this as a wheelchair accessible route.

Sam is correct. The RER D between Gare du Nord and Gare de Lyon has elevator and wheelchair access. It´s only 2 stops and should take about 20 minutes if you know where the elevators are. If this is your first time doing this, allow a lot more time.

Alternatively, arrange with the SNCF for car service between the two stations:

Mon chauffeur

I second Mon Chauffeur instead of taking the train between the stations because I think the help with getting the luggage transferred will be useful. Unless the budget cannot cover it, there is something to be said for making things simple and having a person to help you.

Thank you all for your wonderful information. We will probably use a taxi or try the mon chauffeur (I did not know of this service) for the transfer if we decide to route thru Paris. Our other option is to route thru Cologne. We are Seniors, so yes easier and some assistance will help for sure. Thank you all again for your wonderful information and advice.