While the breeding period to the north, in Switzerland, seems to be early May until early July, and I have not found any information for the Naples area.
Also, are they likely to nest a second time?
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I think it would be difficult to answer except people living around or visiting frequently the Naples region as the bird distribution covers the whole central Europe during summer time.
Then my answer will focus more on the italian part.
One interesting reading would be this document from the European Commission. Disclaimer: this document refers to biological aspects of the turtle dove, but the main topic is the pressure of the hunt on that bird.
In page 10 they mention that:
Stable populations are found in central Europe: Italy, Austria, The Czech Republic and Slovenia (BirdLife International 2004a).
Page 14 table shows the period of migration and breeding countries (EU 25 only).
And in page 17, they shows that probably between 200'000 to 400'000 breedings pairs are present in the whole Italy (Estimation by Birdlife International)
Here are some refrences:
As you can see the distribution os quite large and the figures of estimated breeding pairs very large.
You can also check the observations (not only breeding) with that avibase link.
For the probability to nest the same place, I would say it's possible as in general the birds can go to the same place as the previous year but I cannot say more if the European Turtle Dove have a different behaviour or not. If they have a quiet place where there are no disturbance (Hunting, human activities) or predators, they would probably come back to the same place.
Yes they are likely to breed a second time, and sometimes three times. It mostly depends on the complex graph of nourishment that they can come by to breed up to three times in a year.
They have higher chances of breeding 2-3 times in Naples than in Switzerland according to this document: http://www.birds.kz/v2taxon.php?l=en&s=289
The mean first egg date is May the 18th in the UK, and if you compare the average temperature for London and Naples at that time, spring is about 30 days earlier in Naples, and the birds can migrate there from Africa first, so the mean first egg date in Naples is between the 18th of April and 18th May. They probably start nesting very soon after they arrive.
It depends how good the harvests are and how disruptive the industrial farming is on their migratory path that year, while the nesting time was measured for may in the UK one year, a satellite tracking of a turtle dove placed it's arrival in the UK in June another year.
The result is an erratic timing from April to June in Italy.