I'm not sure a tariff war is in anyone's interest so I sincerely hope that doesn't happen. Regarding the trade deals, the Economist had an interesting piece this week on the personnel required to do such deals, as we have been able to do without such roles for a few decades now. Long story short, we might be able to pull some British nationals back from the EU trade teams, and their counterparts in the WTO, but that alone still leaves us quite short. We could import them from other countries, but I'm not sure that's ideal either, whether from a practical or political perspective.
To put things into perspective though, BIS are advertising for around 500 such negotiators so it's a huge undertaking, and that's just the trade side of things. There will also be a huge job unpicking what is currently EU and will have to be UK from now on, whether that's regulations or laws. Really is a huge undertaking, and I'd be interested to see just how big the budget for the 'Brexit department' will be.