The Egyptians kept their cats to themselves and under close guard. They made their export illegal. This restricted the spread of the newly domesticated wildcat to other parts of the world.
In 500 BC cats were almost unheard of in Greece and Rome. At that time ferrets were the animals chosen for rodent control. Even until 400 AD the cat was rare in Rome. At that time Palladius, a Roman writer, recommended that cats were a good alternative to ferrets for harrowing away moles from artichoke beds.
The earliest record of a domestic cat in Greece is a marble bas-relief picture of 500 BC of a cat on a leash facing a dog.
Once the Romans started to live with domestic cats the spread of this pet commenced in earnest. What lead to the spread of the cat throughout Europe and beyond? There are two theories. The first is that Christianity arrived in Egypt which apparently released the restrictions on the movement of cats. This lead to them being exported to Rome and thence to Europe as a companion to the travelling Roman armies.
The other theory is that the cat’s spread throughout Europe was due to the spread of the brown rat and house mouse. We know that the cat was introduced into Britain (the UK) by the Romans in around 200 AD as I recall.