Will Brexit mean that Britain can start making its own champagne, feta, prosciutto etc?

This is possible but unlikely. 

The EU Protected Designation of Origin (PDO) scheme is part of a wider effort to protect intellectual property across borders; in this case, the scheme is about enforcing the naming rights for a range of geographical indications. Once outside the EU, the UK would in principle not be bound by the EU’s PDO; however, it would of course continue to be a World Trade Organization member and geographical indications could still be protected at the multilateral level through the WTO Agreement on Trade-Related Intellectual Property Rights (TRIPS). 

That said, I don’t think it would be in the UK’s best interest to be outside the PDO, for two reasons. The first is that the legal protection afforded by the PDO cuts both ways, so if the UK had the chance to bottle its sparkling wine as champagne, then everybody in Europe who mixes a brownish liquid could call it Highland Malt Scotch whisky. Since the EU is a larger area, there are probably more products from Greece and France and other countries that benefit from the scheme than there are British products. 

"The legal protection afforded by the PDO cuts both ways, so if the UK had the chance to bottle its sparkling wine as champagne, then everybody in Europe who mixes a brownish liquid could call it Highland Malt Scotch whisky."

At the same time, for the (few) British products that enjoy protection as a geographical indication, the EU is a much larger market. As the Italian minister recently said to Boris Johnson, the EU sells a relatively tiny amount of their products into the UK but the UK risks access to 27 other countries for their products. That’s the fundamental asymmetry. Whatever UK products currently benefit from that scheme, it will potentially cease to benefit from a huge market. 

The second reason is that we don’t quite know what the future UK-EU trade relations are going to look like, but the official government line is that the UK will want to seek as much market access as it can. They want access to the Single Market and that entails playing by the rules of the game. One of these options is membership in the European Economic Area, and membership would automatically entail full acceptance of the trade rules, which include the EU PDO scheme. It will almost certainly continue to apply, so I wouldn’t bet that two years down the road you’ll be able to make your own feta.