I am heading that way in a couple of weeks with a group on road bikes to meet up with our trail bikes that are travelling in the back of a big van.
Would be interested in your viewpoint of riding to Auvergne in a day from the chunnel - it is def a full days riding
but would like to know if you would also opt for the motorways or have an alternative more interesting route in mind?
(1) Where in the Auvergne? It's a big area; from top to bottom would be at least a couple of hours ride on its own. Let's guess at Clermont Ferrand as it is in the middle.
(2) What time do you intend to start off in the morning / what time do you intend to finish at your hotel? The further in time these two are apart will increase your
potential travelling distance.
(3) Stopping en-route? We had two stops of about 30 minutes for coffee and about an hour for lunch on each day, plus two quickish fuel stops. We probably
averaged 45 mph all day on reasonably demanding D roads, so those three simple stops cost us 90 miles. If you stop more or for longer you will lose more miles. You may ride faster or slower of course, or stop more or less, which will add or subtract from your average speed. Who knows?
(4) Work out (1) (2) and (3). Look at the Tracklogs and look at a decent map (see sticky). Yes, you can do it in a long day for sure, probably doing a good mixture of motorway*, National (N) roads and some majorish Departmental (D) roads. How, is difficult for someone else to answer. As Wessie points out, as you appear to be in Hampshire it
may be better to travel over from Portsmouth overnight and crack off early in the morning? But, if the rest of your party is travelling on the Chunnel (or you really need to start from there) then you are stuck with that, I guess? Can you cross overnight, or cross the evening before on the chunnel? If so, you will save one hour due to the time difference.
*There is no shame in motorway miles to cover distance, particualrly if you need to be somehwere by a certain time or to achieve some other purpose, or it is really lashing down with rain or the main roads simply run parallel for miles and miles and miles. That is why there were invented. The purpose of your holiday is I guess to ride off-road bikes in the Auvergne and you want to get there in a day from northern France? Start to cut your cloth accordingly. Sit down with your mates and cobble something together.
PS Michelin's Bis 'tourist' routes map is good. So (often) are the Bis 'tourist' roads themselves.