I have a connection in terminal 3 heathrow to Vienna. I am flying into Heathrow from USA. I know customs and passport control are not needed. It's on a Thursday at 1pm that I get into Heathrow. How bad is the security checkpoint in terminal 3? I have 1hr and 15mins from when my plane lands to get to my connection.
Heathrow has a very cool 'flight connections' page which you can plug in your flight details and find out everything you need to know. I can actually take a guess at what flights you're on based on the information you provide but I'll not post a link directly to that since you didn't quote the flight numbers and you may not want to share that.
But when you plug in your flights you'll see it says:
Time needed for this connection: 70 minutes. You have 1hr 15min available. Our step-by-step guide shows you how to make your Flight Connection.
So, you basically have 5 extra minutes. I'm assuming you're booked on a single ticket and will have your boarding pass for the next leg. As long as your inbound flight isn't very late you should be fine. I can't find any indications of wait times for security in terminal 3 but, in my experience, the wait time for connection security is not that long. You can speed it up by making sure you know the applicable security rules to avoid delays.
In general the minimum connection times allow enough time as long as everything else is going to plan.
If your plane is late the airline will make sure you get on a flight (again assuming this is all a single ticket), it's possible that they'll allow you to use the fast track for security or they'll put you on the next available plane -- there's at least one later flight to Vienna and several flights a day so it shouldn't be that bad.
If you have to stay overnight then your airline should pay for the hotel / food, although that may be an issue if you can't enter the UK but there are rooms on the airside part of the terminal.
Edited to add
If you have issues walking then Heathrow offers special assistance but note you are recommended to request this at least 48 hours in advance. Although they'll still try to help if you request it with less notice. Note that they don't guarantee to get you there on time but will assist.
You need to contact your airline who can arrange the assistance, they'll also be able to advise if you still have enough time to make the connection. You can try contacting the service provider (Omniserve) directly on LHR_feedback@airservcorp.com or call +44 (0) 844 335 1801.
There's some more information on this page from Heathrow Airport Guide (that's not an official site).
I still think, barring delays on the incoming flight, you'll probably have no real problem. Terminal 3 is not that big and security shouldn't be that bad.