Tight Train Connection??

You can take the RER D from Gare du Nord to Gare de Lyon in less than 20 minutes. So it's doable, as long as you hustle (you'll still have to walk to the RER, buy your tickets, then at the other end walk to the train station and to your train.) A taxi could take longer, depending on traffic, but would get you door to door, and luggage would likely not be an issue.

What's your comfort level with hauling arse with luggage in tow if something should happen that delays your arrival from London to Paris Gare du Nord, or you face a particularly long line of people waiting to get a taxi in front of Gare du Nord? It takes about 25 minutes for a car to get from Gare du Nord to Gare Lyon. Let's just pretend your train from London is slightly late and arrives at 7:30, it takes you 5 minutes to get off that train and get you and your luggage from the platform to the line for the taxis, it's a 10 minute wait to get to the front of the line, your cabbie makes it over to Gare de Lyon in 25 minutes - it's now 8:10PM as you pay your driver.

Yes, you can take public transit between the 2 stations, but again, with the stress and luggage, I think you'd be better off just paying the 40 Euro penalty for each of you and just chalk this one up to lesson learned - carefully review all booking details before you click pay.

I have done that many times. That is tight, too tight for me. I normally allow 90 minutes, longer northbound because of Eurostar check-in.
I would change your booking. I am surprised a rail website gave this as an option.

It is 2 stops on the RER (express métro). This should be quicker than a taxi.
I usually do it in 30-35 minutes, from getting off the Eurostar to standing on the concourse at Gare de Lyon looking at the monitor to find my train. But, I have done it many times and know exactly where to go.
You also need to buy a métro ticket from the ticket machines (€1.90), and you don't want to get stuck behind some first-time tourist who can't work out how to use the machine ☺

This video shows exactly how it works: https://www.youtube.com/watch?v=RSAHTKVBVwg

Gare de Lyon is slightly confusing. It has two halves "Hall 1" and "Hall 2". Hall 1 has platforms with letters, A to N. Hall 2 has platforms 5 to 23, odd numbers only. Don't ask me why, its French. When you arrive, the videos will show "Hall 1" or "Hall 2". Go to the correct part of the station and wait. 15-20 minutes before departure the videos will show the platform number (or letter), you can then go and find your train.

You can get a carnet of t+ Métro tickets either in the bar car of the Eurostar or at the tourist information desk near the Caffé Nero coffee bar in the waiting room while waiting to go up to your Eurostar train in London.

That would save some time on that most cramped journey.

Of course unless you are returning to Paris you will have a few redundant Métro tickets. They never expire though.....

I do agree that that is tighter than I would like. I'd probably be able to make it but I know the route down cold. Still too tight really.