My mate in the UK owns a car. He drives it to Switzerland and parks it in my garage. I buy the car from him... i.e. I give him some cash, he takes off the number plates and he/we complete the UK/DVLC 'export' paperwork, he gives me the keys and takes a plane home.
It's really quite simple (to get the form) - easiest at a border checkpoint as they take payment on the spot and stamp the form for you. That said, any customs office (I assume there is one in Basel as well) will do this.
With that, you have a year to complete the import proceedure (i.e. pay taxes, duties, provide the car manufacturers certificate of conformity, etc).
How does it with with the CoC and brexit? Now that the UK is no longer EU, does that mean that you need to get the import inspection which costs a few thousand? Or as long as it's an EU model (LHD?) and already homologated you can get that from the dealer at a reduced cost?
I imported a car from Germany, I think it was 1995 or 96. I did a quick border turnaround from Basel to Germany and back to request the export stamp. One of the Swiss border guards took an instant dislike to me and raised a fuss resulting in a refusal from the German side to give me the export stamp. I telephoned the Zollamt in Bern the next day and explained the situation. They said "no problem" and sent me some forms to fill out.
It was a BMW with Canadian specs and therefore had to have European headlights installed and go trough the process of getting it tested for Switzerland. I brought it to a BMW dealer who took care of getting it registered for me.
It had to be tested for emissions and go through the MFK procedure but was exempted for noise testing because the car's type certificate was recognized.
I don't remember what I had to pay, but it was reasonable and the whole process was painless.