I hope all goes smoothly.
We have adopted four Greek dogs in our family, and it is a doddle . The animal rescue use a local vet to test for everything imaginable and beyond. Obviously leishmaniasis is the important one in the Med , as well as rabies and parasites. All neutering, vaccines, worming and flea treatment is done prior to travel date. They are used to the extreme rigour of UK border controls. Pet passport is all part of the price.
Everything was done for us, including the travel arrangements. Ours came door to door by an ex-pat English courier, but they could have been flown to Heathrow or Manchester. Our total cost was £250 for the animal charity and vet , plus £250 for the van or £350 for a flight . We had nothing to do!
I would not attempt to adopt a foreign dog on my own! Anyone who falls in love with a stray abroad should get it organised by the local rescue group. Also be aware that in Greece , a dog can't be taken without a six week wait to establish whether it has an owner. That gives time for the vet procedures, as described by Mari, to be done. Rabies needs 3 weeks, and neutering.