The solution here could not be simpler, 

* Just hire a car in the UK (pick it up and drop it off anywhere you want), and

* Just hire a car in Europe (pick it up and drop it off anywhere you want).

(It's trivial to transfer between say Folkestone and Calais by train, or between any UK airport and say Lyon.)

It's completely commonplace to do this.

You'd be one of a zillion tourists doing it this year!

The idea of using *one car* for both UK/Europe is a non-starter, just forget about that.

One of the great joys in life is motoring around the Continent.  Enjoy!

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Some further points:

1. Once you are on the continent save a massive amount of money by making a **loop** rather than a one-way rental.

https://travel.stackexchange.com/a/38154/19233

![enter image description here][1]

One-way car rental in Europe is fantastic, but it is incredibly expensive.

2. Whatever you do, in the name of goodness avoid the "living hell" major airports (all London airports, Paris).  Fly to a city such as Lyon or Munich  The quality of your holiday is boosted to a whole different level.

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For the record, note that it is no big deal to take a UK rental car to the Continent for a **day trip** returning to the UK ... if for some reason you want to do that.

(Example, Sixt *promoting* the concept of a day trip! [link][2] )

Again, you would never do this for a longer tour: simply rent a car on each landmass.  Enjoy!


  [1]: https://i.sstatic.net/h8b5a.jpg
  [2]: https://www.sixt.co.uk/one-way-car-hire/london-to-paris/