The European Union does not have a trade deal with Japan (https://en.wikipedia.org/wiki/European_Union_free_trade_agre...).

There is no free trade agreement yet. There are, however, a number of agreements that cover bureaucracy regarding import and export, as well as other relationships. Maybe most of the UK's non-EU import regulations are negotiated partner for partner by the UK government. In that case, nothing would change. But I doubt that ongoing relationships between global partners and the EU do not have any effect on the UK. See http://ec.europa.eu/trade/policy/countries-and-regions/count... for an overview.

Maybe it's possible for the UK to get custom tailored agreements that are more beneficial. But I'm afraid that until all of that is settled, a lot of time will have passed. Negotiation about the FTA between EU and Japan has been going on for about four years now. Other fora of exchange have been established decades ago. Until the UK can benefit from their own negotiation efforts, the EU might look more attractive to Japanese investments and trade.

International trade is complicated and takes time but the Leave campaign made it look like everything's going to be fine in just a few years.

Yes, we will probably remain with the EU for long time coming. It will be a slow gradual process.

Hopefully, since that would have the least negative fallout. The untangling of connections across the Channel will take some time.

OTOH, looking at other possible referendums across Europe, I would like it to be clear what an exit from the EU entails as soon as possible. I don't think leave voters will get what they had hoped for, vide https://amp.twimg.com/v/6ca5195b-a8a5-4b20-b209-92440b9a25d6