Car tax question: Bought a used car without tax, can I take it home and then get it taxed or do I have to tax it first before taking it home?

In the UK, you tax it before you take it home.
You can do this online, by phone, or at a Post Office.
See https://www.gov.uk/vehicle-tax and the section headed "Buying a vehicle" on https://www.gov.uk/government/news/vehicle-tax-changes
Note that you will need to have insurance cover in place before you can tax it.

You pay the tax when you buy the tags in my State. So that would mean you do not have to pay it until the temporary tag the dealer put on your car expires. Normally this is 30-days.

You could take the risk or you can bring your mobile phone and pay for it there and then once you have bought it, via the internet.

You must tax it first which you can do online. It is then taxed immediately.
UK

If you drive without tax you risk getting caught by a police ANPR camera. I would tax it first. But I am not you.

It depends heavily on where in the world you live. In much of the US there is no tax paid on private purchase used cars.

You have to pay the tax to register it and get plates. If you bought it outof state, you might be able to get temporary in-transit tags