Do I need a UK visa to leave the airport during a layover?

The OP is will be in transit to the US.

This UK govt page contains the following text:

Transiting without a visa

You might be eligible for ‘transit without visa’ if:

• you arrive and depart by air
• have a confirmed onward flight that leaves on the day you arrive or before midnight on the day after you arrive
• have the right documents for your destination (eg a visa for that country)

One of the following must also apply:

• you’re travelling to (or on part of a reasonable journey to) Australia, Canada, New Zealand or the USA and have a valid visa for that country

(Many other non-relevant situations omitted)

• you have a valid USA permanent residence card issued by the USA on or after 21 April 1998

(Many other non-relevant situations omitted)

It appears that the OP meets the stated qualifications and may transit without visa.

Still, the above page also says that entry into the UK is up to the Border Force officer:

You won’t be able to transit without a visa if a Border Force officer decides you don’t qualify under the immigration rules. You can apply for a transit visa before you travel if you’re unsure whether you qualify for transiting without a visa.

There's also the possibility that a sloppy airline employee may misread Timatic, and refuse boarding in Vienna. If the OP chooses to approach this in a very conservative manner, he may wish to apply for a "visitor in transit" visa to increase his chances of being allowed to board and to enter.

Start at this UK GOV page, which broadly describes the transit visa issue. The OP will be going through UK border control because he'll stay overnight in London. This text appears:

You’re going through UK border control

Apply for a Visitor in Transit visa if you’ll be going through UK border control but leaving the UK ok within 48 hours.

Following the Visitor in Transit Visa link, one is taken to Visitor in Transit Visa page, where this text appears:

Documents you need

To apply for a Visitor in Transit visa, you must have a current passport or other valid travel identification document.

You need to provide evidence that you’re allowed to enter the country you’re travelling to, such as a:

• residence permit
• green card
• valid visa
• If you’re not a resident or national of the country you’re travelling to, you may need to explain why you’re going there. You may need to provide details of where you’re staying.

You must also provide evidence that your onward journey is booked or confirmed, such as:

• a flight booking email
• printed tickets
• confirmation from a travel agent

Your onward flight must be within 48 hours of your arrival in the UK.

Bring your visa and documents with you when you travel through the UK.

Applications from outside the UK must be made online. The text continues:

Apply outside the UK

You must apply online for a Visitor in Transit visa.

You’ll need to have your fingerprints and photograph (known as ‘biometric information’) taken at a visa application centre as part of your application.

You may be able to get your visa faster or other services depending on what country you’re in - check with your visa application centre.

This page has an Apply Online link to begin the application.