Lots of misunderstanding of the WTO situation here.
it's entirely between the EU and the UK and there's no third party or outside body which has to be involved or considered.
No! That is what the WTO exists for - to regulate international trade. The WTO has strict rules about trading blocs. The EU cannot continue to treat the UK as if it is a member once it has left. The Most Favoured Nation principle underpins the WTO regulations - you cannot trade preferentially with one country over another unless you have a proper trade deal. The UK loses all its preferential access to the single market.
As for 0% tariffs, Patrick Minford and his Economists for Free Trade have suggested in their highly dubious and widely discredited model that the UK can continue to trade easily by dropping all tariffs to zero percent. However his model assumes that all other countries will reciprocate this for us, which is actually highly unlikely! What's in it for them?
WTO schedules are not yet set up for the UK - several countries have objected to allowing us to roll over our current EU schedule (Australia,, New Zealand, Brazil and of course Argentina, who are not great pals of the UK).
Many countries will be keen to trade with the UK after Brexit but it will not be essential for them to do that deal immediately as they will carry on trading with the rest of the world (unless they are selling a product that is entirely dominated by the UK market - Marmite?!). They can take their time to negotiate a beneficial deal. It will be essential for the UK to make deals as soon as possible. Essentially, we will be desperate. This does not put is in a great bargaining position!