No, you don't go through immigration. You will not enter the UK/Ireland common travel area.

we just flew from boston to barcelona and back from madrid to boston through heathrow. Each flight arrived and took off from terminal 5, but required us to go through a passport check (quick) and security (also pretty quick). However, it took about 45 minutes to an hour to travel to the gate, as we had to board a train and go back to the main terminal each time. Our last flight (which we barely made despite a 2 hr layover) was just yards from where we landed, but it took us an hour to get there. Crazy system- I will avoid heathrow if at all possible from now on.

Also as we are flying Business class we are hoping LHR has a security fast track lane? Does one exist at Terminal 3? ( I think we fly into T1 and out of T3). Any words of wisdom? Many thanks.

Yes, but you need to be sure you get a fast track pass from the cabin crew on the VIE-LHR leg. On intra-Europe shorthaul flights sometimes (often?) the FAs don't pass them out, so you should ask. You may be able to use fast track just by showing your business-class boarding pass. But "airside" (i.e. inside security) transfers at LHR are generally quite efficient anyway, so if you don't get one, don't despair. What airlines are involved?

Several years ago, I flew a mileage run to / from the U.S. to LHR to maintain my AA Platinum status.

I was required to go back through the line that included passport control and a security search.

AA wouldn't allow me in to the Admirals Club unless I went through this process. I thought it was strange, but I didn't seem to have a choice.

We are flying Austrian Air into Heathrow from Austria and then Air Canada from Heathrow to Toronto. While we have a 3 hour layover I am still concerned as any flight delay from Austria could make the layover much tighter. With all the staff shortness.causing delays up to a couple of hours at Heathrow I am a bit concerned. Thanks for the heads up that we may need to request the Fast Track passes.

With all the staff shortness.causing delays up to a couple of hours at Heathrow I am a bit concerned.

My understanding is that those delays relate to border control, so should be a non-issue.

3 hours is more than fine, even without fast track.

Just the slightest bit of bad weather sends Heathrow into some sort of fugue state, so even if your flight is late landing I think 3 hours to do a T1 > T3 connection is ample. As travelgourmet says, the big delays are at passport control, which you won't experience.

Terminal 5 at Heathrow when it opened a couple of years ago introduced some new security measures including showing your passport and getting some sort of entry card (for the terminal) stuck in your passport as yu pass through the various controls. I went through this once and don't know if they continue to have all these controls within terminal 5.

As far as the bigger picture, when you depart Italy for Vienna it is an intra Schengen flight and so no passport control in Italy. In Austria, you will under go a leaving Schengen passport check and arriving at Heathrow you will not officially enter the UK hence no formal UK border controls (all providing you're travelling on a single ticket and have checked your luggage through). So you will undergo all sorts of security checks but since you are not flying to the United States, you won't be involved with the asinine checks all flights to the USA get at international airports, luckily for you although I hear Canadian immigration officials are pretty rough also but that won't come up till you've crossed the pond.

Terminal 5 at Heathrow when it opened a couple of years ago introduced some new security measures including showing your passport and getting some sort of entry card (for the terminal) stuck in your passport as yu pass through the various controls. I went through this once and don't know if they continue to have all these controls within terminal 5.

I have not seen any sort of entry card at T5. Perhaps they had them and quickly got rid of them, but they certainly did not have them last month when I connected through there, nor any time I have connected through there.

All that you will need to do is clear security and head to your next gate. There is a boarding pass check, but this is a non-event in my experience.

The only caveat that is a bit odd about connecting with multiple carriers at Heathrow is with regards to checked baggage. Some carriers will request that you stop at the service desk (after you clear security for your onward flight) and present your luggage tags. They use this information to make sure they "pull" the baggage from the inbound carrier to your outbound flight. Again, this is usually a non-event and I have no idea if it is really necessary, but it is something to be aware of.