The Times is similar to the BBC.
It isn't the organisation that people grew up with and its only real hold onto conservativism has been economic.
Socially it is much more in the centre, even left of centre on some issues.
Murdoch is very much aligned with the money world and whatever that establishment wants is the line his media takes. All the money/power really cares about is protecting its wealth. It will go left on social issues if it thinks that ensures that...if that starts to become unpopular as 'woke' has recently it will pretend it never pushed anything woke.....It will essentially do anything and become anything.