After the completion of the Brexit, the trade between the EU and UK will likely increase in complexity and create new risks and trade barriers. Although it is still too early to know, any of the following scenarios could be possible:
The UK status would change from EU member to “third party country”. The UK would then no longer be able to operate under the rules of the treaty on the functioning of the European Union. Instead, they could only rely on the rights based on their WTO membership. That would mean the EU would then automatically execute customs formalities similar to those in effect towards other third party countries.
Free Trade agreements between UK and EU could certainly help to overcome the fact that duties would have to be paid, but those would only cover and create benefits for the transportation of origin goods of both contracting parties. The consequence of this would be that deliveries would need to be accompanied by a proof of origin, an additional process. A loss of the customs tariff preference of the UK towards the EU could also be a result.
To cover non-origin goods the creation of a customs union between UK and EU could be a possible solution. A requirement for such a union would be the acceptance and adoption of the common EU customs tariff by the UK (even though the UK has no influence on its development). Nevertheless customs formalities would need to be executed. For example, in the case of duty unpaid goods which could be handled via the New Computerized Transit System (NCTS), there would now have to be a corresponding guarantee between the partners in the transaction itself, if the UK affiliates the common transit simplifications.
As the EU Customs legislation would not apply for the UK anymore, the UK would need to create their own customs code or tariff. Alternatively, the UK could adopt the EU Customs Code, as well as establishing a common customs tariff.
Exporters from the UK would need to apply for an indirect representation in the EU in order to issue a Custom declaration.
Beyond these Customs issues, conventions in regard to safety and security regulations would need to be arranged, possibly similar ones as with Switzerland or other States. Otherwise summary declarations within given time limitations would be required for import - and export. Depending on the outcome and direction, it is also likely that the US TSA will need to review/revise its Aviation Security Mutual Recognition cargo agreement with the EU, which currently includes the UK.
These of course are preliminary thoughts, and only reflect on some parts of the issues trade might face as this unfolds. We are far away from understanding all of the possible permutations and challenges.