I did this in February. It was not the busy season, so it was fairly quick. End of June will be busier, all the steps might take longer. Is there a train after this one, just in case? There are some train apps that will list all the trains and you can buy tickets through them. I didn't use to buy but I liked the listing of all the trains.

I did check my bag from LHR to BLQ and my baggage was out within 10 min. A cab was right outside the airport. I had to stop at an ATM to get euros - which was about an extra 5 min and the driver knew a place on the way. It only took about 10 min to get to the train station.

However, are you familiar with buying train tickets? Do you speak Italian? Will you buy your train ticket prior or do you need to buy it there? I don't know if the windows are open that late. If they are, you may wait for a long time. People do not move fast. There are a few kiosks that you can put in euros or your credit card. Screen is in Italian and it was not obvious to me if it could be translated to another language. That visit I did the window. Luckily, I had some time. This last visit, an employee was standing at the kiosks and helping people get tickets. My credit card didn't work - I had to have my PIN which I did not have at the moment. I had euros so I paid with that. She selected a train that was to leave in about 5 min - but it was delayed by about 17 min (the strike helped out on that one.) You may have to walk a ways/escalator to get downstairs, so allow about 5 min for that, more if you have others with you, lots of luggage or mobility issues.

If you buy a train ticket that can be used on any day/time (local trains), it has to be stamped. If it is not stamped, a conductor might check tickets and you could get a huge fine on the spot.
If you buy a ticket for a specific date and time, you don't need to as it can only be used for that specific trip.