Not sure when Gordon Brown was "one of the Thatcher chancellors" but he removed mortgage tax relief in 2000, house prices have risen steadily since, did you read up on that one before you posted it?
here is a guardian article discussing it.
https://www.theguardian.com/uk/1999/mar/10/budget1999.budget3
In 2000, mortage interest relief was costing the government £1.4 billion, so it was equivalent to a mortgage subsidy this big. But interest rates were falling at the same time, cutting peoples mortgages and compensating for its abolition. In 1990 the bill was £7.7bn. The conservatives had also been whittling away at it in stages since then, but in 87/88 they announced a halving of the available amount to new buyers. It caused a boom in puchases at the old rate before it was abolished and a crash at the new rate afterwards.
Tax relief on home purchase loans was a subsidy for the rich to help them buy homes. Thatcher was quite the socialist in some ways in presiding over its removal by stages. There was no housing shortage when she came to office. She would not have liked this shortage at all, because she would have seen it as harmfull to the tory party. Curiously, when May talked about helping the JAM's it was Thatcher rhetoric, but no action.
Wealth was increasing fast in 2000 under labour whereas both under Tory Thatcher and now under Tory Cameron/May it has frankly been shrinking away. Thatcher made the mistake of having a guillotine process, whereby if you bought a home one day it would continue to qualify for tax relief. If you bought the next day it would not. This accentuated the problem and showed precisely how much of a boost to prices that subsidy gave. The market in action demonstrated the difference in price of a home with or without tax relief.
The Thatcher strategy was to make people home owners, because then they would feel wealthier and vote for the party of the wealthy class. Allowing prices to rise and rise because there is a huge shortage of homes means that people see they no longer have the opportunity to buy a home and will never get the benefits of doing so. They will be better off not voting for the party of the wealthy, because it isnt going to be them.
Thatcher succeeded mostly because the nation was tired of union misbehaviour, and because the economy was collapsing. Though in fact the collapse of the world economy at that time was due to the oil price shock - oil producing countries successfully created a cartel and demanded vast price increases for the oil they were selling the world. It was not unlike what is happening today therough globalisation, in that the poorer countries suddenly captured a bigger share of the world's wealth. Inevitably this meant a drop in living standards for developed countries, and social unrest. (Brexit will do the same)
New labour was a reaction to the tory success, creating a political party much more to the right, favouring the wealthy, or at least those who imagined they were wealthy or would become so. Traditional Labour is going to succeed now because society understands wealth inequality means a big chunk are losing out. It is a generational thing. Those who lived under Thatcher saw the benefit of the giveaway of national assets the tories engaged in and older people are still voting tory. The young, traditionally poor as they start out, see no benefits to be had because the government will not give them a free house or free shares in former nationally owned companies. Instead it gives them a stonking debt for education, which in reality is an investment for the whole nation.
The liberals trialled Corbyn's policies in 2010 with great success. Yet having been voted in by youth demanding a share of national wealth, they then supported the tories in doing the opposite. They promised what labour will deliver. So of course people under 50 voted labour at the last election.
Housing has been a core driver of who gets to be government for 50 years, maybe longer, and the right have cocked it up now.
Brexit will shrink the Uk economy and make everything worse. If it happens, it will discredit the tories who have pushed the policy. It will make the need for a more socialist state all the more urgent, and more people will be in the camp of the 'have nots'. Brexit will cement in labour wins, and might even restore the fortunes of the liberals, as the new right wing but pro EU party. The conservatives risk turning into new UKIP on 10% support.