Two countries and transport

fly in to one country, travel to the other and fly from there

that's the most efficient approach

Without knowing where in the Uk you’d really like to go or where you’ve been before, this request is a bit tough

But, what about, fly into Gatwick then train to Brighton and bus to stay somewhere in the S Downs National park which has some good hiking (Seven Sisters, Beachy Head), Afterwards take the ferry from Newhaven to Dieppe (I haven’t checked that they take foot passengers but am guessing they do), then see Normandy and then travel on to Parts and fly back from there.

Another option if you're interested in DDay history is to head down to Portsmouth and visit the excellent DDay museum there, which explains the whole preparation and staging process involving the invasion armada from ports along the south coast of England, then maybe hop on the ferry over to Ouistreham (port of Caen) in order to visit the Normandy sites, eventually catching the train back into Paris for your flight home.
An open jaw (multi-destination) flight plan of flying into London and home from Paris would streamline your itinerary and likely save a day or two of back-tracking I think.

Yes the Newhaven route still takes foot passengers, even if the trains no longer stop at the side of the ship in Dieppe as they used to. Now you have to go to Dieppe Ville station for the train, but much easier than at Calais.
The old Dieppe boat trains were some of the most comfortable train cars SNCF owned- absolutely wonderful to step off the boat at around 3am and straight onto the waiting train- literally armchairs you sank into in compartment style train cars. Close the curtains, switch off the compartment lights and sleep through to Paris.
Thanks for the memories.
Now it's just normal regional multiple units.

I'll share my personal experience. Flight from USA to Europe, including LHR, round-trip is always cheaper. For open jaw, ie USA to LHR, CDG to USA, flights are somehow more expensive. If you use award ticket, that maybe okay depending on whether you can grab one. Note that department tax from LHR is higher as comparing to other hubs in Europe. Fly USA to LHR, spend time in London where public transport is super easy and convenient, train Eurostar to Paris, or ferry Dover to Calais, trains in France, fly CDG to USA.

Just to chat about the public transport part. This is completely doable if you are willing to take day tours once you are at Normandy. We found Overlord tours out of Bayeux excellent. The further you purchase train tickets ahead, especially foe Eurostar, the better the prices.

How lomg are you allowing for this?

What sort of things are you hoping to do and see?

Take a day tour of Normandy.
Don't miss some of the great smaller cities and towns in the UK:
York
Cambridge
Winchester
Bath
Stratford Upon Avon
Salisbury
Canterbury
Windsor Castle