Food is already more expensive than it would've been if the referendum vote had gone the other way, largely because of the fall in the £. Most people tend not to notice because the way it becomes more expensive is with shops gradually putting up prices on individual goods a bit at a time, and the connection with Brexit isn't readily apparent, but the causal effect is definitely there. If we leave without a deal, then the value of the £ is likely to fall even more, and prices will correspondingly go up more, albeit - unless shortages cause a sudden spike - there may be a time lag of a few months because of competition pressures on supermarkets: No company likes to suddenly jack up all its prices at the same time. Any sudden imposition of tariffs will have a big impact too.
If on the other hand we leave with the May deal (very unlikely but possible) then I suspect prices won't rise much more than they already have, at least in the short term, because the expectation of the UK leaving with some sort of deal would have already been accounted for in the exchange rates to some extent.
People, especially on the leave side, are making an assumption that we'll suddenly be able to import food more cheaply from other countries. In the long run that might be possible, but it brings up issues of food safety and for other free trade deals to be approved - which takes years. Also, although people are talking about importing stuff by ship across the oceans, a lot of food is too perishable for that anyway.
Besides, another thing that noone here seems to have keyed into is that the EU already has arrangements to allow many of the poorest countries (including for example, much of Africa) to export to the EU tariff-free. The purpose of those arrangements tends to be for the EU to play good citizen and help the very poorest countries, but undoubtedly that helps the prices of some raw commodities (for example, sugar) within the EU too. And obviously, if we leave the EU without a deal, we lose that.
The UK by the way is nowhere near self-sufficient in food production, so there's no way we can avoid having to import large amounts of food for many years into the future. Brexit or no Brexit.
In conclusion: No-deal Brexit = substantial increases in food prices over time. May-deal Brexit = maybe some increases but not too much. No Brexit = food prices drop back to roughly what they were a couple of years ago.