No. You will need to purchase reservations for most if not all of that journey.

If you are a UK resident, you need to buy an Interrail pass. With an Interrail pass you can travel on a maximum of two travel days in your country of residence. 

Depending on your age (may determine if the pass is cheaper) and if you are willing t book long advance (but you have to do that anyway for the RES) for a return along this route advance tickets may work out better-and a little cheaper too. Check-as always-seat61.com for a takeyoubythehand complete description.

Esp for the BRits it seems an eternal enigma to read and then check to find they also have to use INTERrail -EurRO rail does not even exist, its EUrail for those out of the whole continent+assorted islands to it

Yes you have to add reservation fees to the cost of your pass and for us Brits all the expensive ones are where we intuitively want to go - Eurostar, TGVs, Thalys and many Cross border trains..

To get to Venice you could easily add up to 100 euro per person in those fees

If you want alternatives you could save a couple of travel days by getting a cheap flight to virtually anywhere over the channel and starting from the nearest station. I have used Catania for a run up from Sicily and will soon fly to Nice, then train to Venice, through the alps to Vienna and finally fly back from Prague.

Also if you are not bothered about Paris you could try E* to Brussels, ICE direct to Frankfurt. Then on to Munich or Vienna, then through the alps to Verona or Venice (Or come back that way).