You have that the wrong way round. The cost is collected by the firm that receives the goods, so in your example it is Tesco who will have to pay. The EU based export firm are not involved although there will be a requirement for more paperwork but that is it for them. The onerous task is on the UK government to track what comes in, calculate the tariff and serve demand for payment on the importer. It can only logically push up prices in the UK.
As for non EU trade - when we leave the EU we will have worse trade deals with non EU countries than we have now so we will be hitting these suppliers with increased tariffs at the same time. Most of our food comes in with zero tariff via the EU trade deals (see image), we lose all of these in Jan 2021. While brexiteers like to talk this up as an opportunity it is in reality a battle to get back to where we are.