How many times can American Tourist re-enter UK in same 6 month period?

If in the middle of my trip, I fly to Italy for a few days, and want to fly back to UK for a few more days before heading home, can I re-enter the UK?

Yes, though, as always, the immigration officer can deny entry if you do not convince him or her that you are a "genuine visitor."

And if I can re-enter in the same 6-month tourist limit - how many times can I re-enter the UK (for example between April-Oct)?

There is no limit. You can reenter the UK as many times as you are able to while convincing the officer each time that you are a genuine visitor. This will become more difficult as the amount of time you've spent in the UK increases. It will also tend to be more difficult if the stated duration of your intended stay is longer or if your plans are more indeterminate.

If you want to stay for more than a couple of months, it's probably best to have your return flight booked when you arrive, have copies of (at least three months' worth of) your bank statements to show that you can support yourself without working, and evidence of your life in the US or other country of residence to show that you have something to return to. Say that you're essentially planning to stay until the date of your return flight, though you may mention the possibility of making some sort trips out and back in the meanwhile. If you're admitted on the first try, you're unlikely to be denied admission on subsequent arrivals between then and the date of your first flight.

The UK has a history of holding people to what they said at the border, so someone who was admitted for six months after saying they were planning to leave in three weeks, who subsequently decides to remain for much longer than three weeks, can be banned from the UK on a future visit on grounds of deception. It is therefore highly inadvisable to base the duration of your stay on the date of departure to Italy and then return after only a few days for another visit. That could easily be seen as an attempt to game the system.

What will make them itchy is if you are trying to

• seek employment in the UK. The #1 thing they look for there is means of support, i.e. They want to know you're not using up all your savings.
• live in the UK, as in, make that your primary residence.
• become a burden to social services, like public housing or healthcare.

Immigration's job is to assume you intend those things, then cheerfully accept evidence to the contrary.

They also like to see you have a plan (which includes not doing any of the above).

What you say when you reenter should match what you say on your first visit, because they remember. They dislike when what you say doesn't match actual facts, because that means they can't trust anything else you say. That makes it hard to accept your evidence.