For people asking about pharma - it's difficult to condense down. But fundamentally, there are several issues - regulatory authorities, licenses and supply chain.
UK authority to license drugs is the Medicines and Healthcare Regulatory Agency, MHRA. In the EU, all the healthcare agencies co-operate to run a central licensing procedure, so you make one application and end up with a licence in each country, via the European Medicines Agency (EMA) which is in London but now relocating to Amsterdam thanks to Brexit. We (UK) have been at the forefront of the EMA, driving standards and leading the science. From next April, no more. Existing drugs will continue to have licenses, but in the future it will take longer to get new drugs as companies will have to submit separate application to the UK and we won't be a priority market compared to EU27.
You can't sell a pharmaceutical product in this country unless you have a license. So, if the NHS runs out of its usual insulin/drug of choice, they an only source it from a company who has already submitted and obtained a licence in this country - showing that the drug is efficacious, safe and a suitable quality. You can't just ring up "drugs r us" and order some from another company in another country if they don't hold a UK license.
Supply chains can be lengthy. The rules for importing drugs into this country are complex, to ensure that they comply with the conditions of their licenses. They will have been tested at the factory when they are made, but there are requirements for testing when imported, and it legally cannot be sold until it has been cleared by a suitably qualified individual. At the moment, thanks to the EMA co-operation, this is done once on entry to EU on behalf of all countries. How this is handled in the future is still unclear - depending on the deal we get, we could retain certain levels of co-operation like mutual recognition of testing. If not, then we'll have to do it all in the UK, and there are limited numbers of suitably qualified individuals.
This is all separate from the concern around cold supply chain like vaccines and insulin - you can hold drugs in quarantine in suitable chillded warehouses while the testing and release is done. I believe most of this kind of shipment is made by air anyway. However, if the Customs guys are saying they don't have the resources to do the import permit paperwork for ANY perishables, whether drugs, food, flowers etc, then unless special arrangements are made, certain drug supplies could be impacted.
Finally, if you want to know whether your drug is made outside the UK, have a look at the carton and/or the leaflet inside the pack. There should be a statement at the end of the leaflet that says: "Marketing Authorisation Holder [ie license holder] Company Name & Address. Manufactured by Company Name & Address".
Happy to answer any other questions...