You mentioned a south east London borough. In somewhere like Lewisham you're looking at around a median price of £800k which will get you a nice semi. Perhaps it's somewhere like Southwark, which is nearer the £1 million mark for houses?
https://www.rightmove.co.uk/properties/164081081#/?channel=RES_BUY
Near where I live, there are many homes just like this that sell for for £180-250k
"They don’t all have (and we certainly won’t have) an extra £12k a year to pay a property tax."
There are two different things being proposed at the moment.
One of them is a replacement for council tax and there is no indication that, if this were to go ahead, then average bills would rise dramatically.
The second is a replacement for stamp duty. In this situation, after a certain date then stamp duty is replaced with an annual tax based on the value of the home in excess of £500k.
One proposal is 0.54% on values from £500k to £1 million and 0.81% on values above £1 million.
So, at the moment, if you buy a house for £450k then you will pay £12,500 upfront for stamp duty. Under the proposed plan there would be nothing to pay ever.
If you buy a house for £600k today then you will pay £20,000 upfront in stamp duty. Under the proposed plan you would then have to pay £540 per year.
In your situation, you mentioned that you are facing a stamp duty bill of almost £60k. This would indicate a purchase price of around £1.15 million.
Under the proposed plan, instead of paying almost £60k upfront you would have to pay £3,510 per year instead.