Quite surprised that a Labour Government is considering this.
For others, that extract is from
here. (paywall but visible via archive websites)
The text is about the £20bn blackhole which the government needs to sort out. The article briefly mentions HS2 to Euston and mentions the possibility of a tax levy similar to the Northern Line Extension to recuperate the construction cost of the tunnels. The full Euston station itself should be able to fund itself via the oversite developments. However, I could see the Treasury handing the construction + funding + operation for X years off to a private company to reduce their risk.
While cancellation isn't impossible the article notes that "
there are fears that cancelling infrastructure schemes aimed at improving Britain’s connectivity will have a detrimental effect on the economy in the long term". Funding the tunnels via debt and paid back via a tax levy is quite possible, debt that funds infrastructure assets has a far more preferable credit rating than debt for day-to-day expenditure.
I think HS2 to Euston isn't at enormous risk with raised taxes and a decrease in the use of hotels for housing asylum seekers being more likely, though I wouldn't be surprised if the Stonehenge tunnel is quietly forgotten about. The Times article doesn't even mention HS2 as one of the key points.
A Treasury-funded HS2 2a is almost certainly off the table, but if the private plan found some financial backing it would likely get the signoff.