"ADDED: If he was from Menton, I wonder why he would travel from Dieppe in France as opposed to somewhere from his own country (the then Kingdom of Sardinia?) or Italy? The distance is very huge between Menton and Dieppe - I checked and it's over 700 miles."
He may well have considered France as "his own country" rather than the Kingdom of Sardinia, just like Garibaldi thought the opposite. (I've travelled by train from Nice to Ventimigilia first station across the Italian border (by accident) and you really wouldn't know you'd crossed a border!! so maybe it wasn't such a big deal for most people only those who were more politically minded.)
There could be a few reasons why he got the boat from Dieppe:
He could have moved to a different part of France initially before moving onto England.
He could have got a boat from Nice? to Dieppe and then another boat to England, although this would be a long journey because you would have go around Spain and Portugal, and the bay of Biscay to get to Dieppe.
Its seems that the railways may have arrived in Nice by the 1850s. There is are a reference to almost 80000 men and 5000 horse being transported by train from Paris, to the "Mediterranean or to the frontiers of the Kingdom of Sardinia" for the Austro-Sardinian War, between 20th and 30th April 1859. That's a lot of people in 10 days, so it would appear that it is reasonable journey time. He could have travelled by train from Nice? or maybe Marseille? to Paris and then onto Dieppe for the crossing to England. I suspect this was the quicker and cheaper method, (other than walking the distance of course).
People were much more mobile, even the poorer people, during this time than we tend to think. They were also travelling to America and Australia, so people probably weren't phased by long journeys, in return for the promise of a better life in England or in America or Australia. You never know, maybe wherever he came from in France, that England was only meant to be a stop off, but ended up staying.
