We have been traveling in Italy 4 times in the past 3 years, from Puglia in the south to the Dolomites in the north. I drink only tea at breakfast, with milk, and have never had trouble getting it just the way I want. And with fresh milk, not that UHT stuff in little packets.
Most hotels will have self-service for coffee and tea. This will come from a machine that offers many options in a picture menu, with one of them usually “hot water” (Acqua caldo if it is only in Italian). If hot water is not an option from the machine, there will be a nearby hot water dispense, near the array of teabags on offer. This is actually preferable, as a long line can form at the coffee machine. If the hotel is popular with British guests, there will usually be teapots to fill, along with the teacups and saucers. I have not seen fresh milk there but go and get it from the cereal area.
Some hotels have breakfast servers take orders for morning beverages as you enter the breakfast room or just after you are seated. This is more common in 4-star and 5-star hotels. If this is the case, just order “black tea” (tè Nero) or “English tea” (but then you might get Earl Grey). I would not ask for milk, but would get it myself from the buffet. They will bring a pot of hot water, a teacup, and teabags, either on a saucer or in a basket with an assortment. Although I am not all that fussy about the brand, I do bring my own teabags as a backup just in case they only offer Earl Grey (which I detest).