Since SDSR, most times that "the defence budget" has been discussed, the costs of Successor have been treated as part of it and not separate, eg
House of Commons Hansard Debates for 14 May 2012 (pt 0001) ,
Trident spending to account for one third of defence budget within a decade - Telegraph , and
http://www.rusi.org/downloads/assets...Term_Blues.pdf
It's up to you if you want to consider the defence budget to have been cut by SDSR and made to include Successor, or cut even more but not made to include Successor. From the Treasury's view, the distinction is irrelevant; they hold funding lines for elements of Successor and funding lines for the rest of defence, and options can be taken against any of them without anyone being concerned over whether it comes from the 'Defence Budget' or not. Similarly, if the entirety of Successor was cancelled tomorrow, the funding would not automatically be shared between the 'rest of defence' (although you can guarantee that the Chiefs would see an opportunity to argue that we needed some more conventional forces).
Another reason that the Treasury isn't interested in the distinction is that the MOD hasn't yet decided what will and won't be considered to be part of the capital cost of Successor anyway.
House of Commons - Defence Committee: Supplementary written evidence from the Ministry of Defence shows that the cost of paying Rolls Royce to develop the PWR3 reactor might not be included, probably since it is expected that this reactor will also be used in the MUFC (the replacement for Astute).