Looks like a Rolls Royce Merlin engine, not a BMW.
Please correct me if I'm wrong but after WWII there were quite a few ME109s available on the cheap for nations such as Israel. Spain had both early and late models and so did Switzerland who had a lot of perfectly good aircraft from all combatant nations that were flown their by their pilots and crew for various reasons. Switzerland interned the crews/pilots and kept the aircraft. Not sure if they utilized them or not.
At the end of the war, the Spanish planes were reconditioned completely and fitted with Rolls Royce engines, supposedly because original engines and parts were in very short supply and it was just cheaper to fit the new RR engines. I don't recall if there was any improvement in performance. The original BMW engines were very good and considered to be some of the best engines of WWII. Extremely expensive and almost impossible to keep up sufficient supplies of completed motors or parts to keep up with demand. This condition persisted after the war as well.
I am not sure about Israel but seem to remember something about them looking for replacement engines for their 109s as well. IIRC, they went to the relatively abundant RR Merlin engines as well.
There were several nations that had ME109s sitting idle on airfields after WWII. Soviet Union, Germany, Norway, Sweden, Finland, some Baltic countries, African countries, Mediterranean countries and even France, Belgium and Denmark. I am sure I missed some.
At the end of the war, no matter what condition these planes were in, just like Allied planes, they were all considered to be a source of scrap metal. Some survived but not many as is obvious now. Who could afford to fly them, let alone keep them in serviceable condition?? Certainly not many governments who were now into jets and felt they were valueless and obsolete. Some of them made it to Korea (Corsairs/P51s) and Viet Nam (Crusaders) but very quickly were replaced by jets and helos.
Again, IIRC the Me109s were rented from the Spanish Air Force.