I think that HS3 (if it goes ahead) would be funded in a similar way to HS2, ie out of it's own budget that exists solely for it. Money available for upgrades on the existing network isn't affected by HS rail construction.
In the current Control Period that is certainly correct. But in the
next Control Period you could be wrong!
The Government, any Government, has only one source of money - taxes. The amount that can be raised, whether through Corporation Taxes, Excise Duties, Stamp Duty, Income Tax, VAT or any other tax is limited by the general economic situation and the politically acceptable level of taxation rates.
This income is not hypothecated and goes into one pot from which the Government, of any political flavour or leaning, makes value judgements and allocates funds to pensions, health care, income support of the lower paid, education, defence, its own running costs, flood defences, transport and a host of other things as it sees fit. At a lower level the individual ministries will allocate funds according to their priorities - in the case of transport a sum will be spent on roads and another on railways and some more on aviation and buses. Some of the money spent on railways will be used for on-going operational spending on the existing railway and enhancements to it and some will be spent on new construction such as HS2. In this sense you are correct - at this level the budgets
are separate, but at a higher level they come out of the same pot.
If money becomes tight, as it might well do over the next few years, experience shows that it is the prestigious, high profile, politically useful projects that get funded and on-going, boring maintenance gets squeezed. Hence pot-holes in the roads - the roads still work, just not as well as they could do.
The initial dance of the bureaucrats is now starting for the allocation of funds for Network Rail's next Control Period starting in 2019. It would be unwise to expect that NR will receive an increase in funds over the current settlement and remember that since NR was categorised as being in the public sector it can no longer raise funds from the capital markets to fill any gaps. It now has to live within the money it is allocated. To keep HS2, and possibly HS3, alive may well mean that the existing railway has to make do with less than it gets at the moment.