Well UK has plans for 7 Astute class submarines and it has 60 billion budget - it is smaller submarine though but unlikely to be that much cheaper than Russian subs anyway
Russia plans 9 submarines from 90 billion budget -rough calculation I know but it seems that the Russian budget can afford it.
It's not that important how big the budget is. What is important is how much is left for aquiring new things.
For example, the U.S. defense budget is around 500 billion, excluding overseas occupation "operations". However 90 billion is set for bying new military plattforms and 60 billion is set for research. So about 150 billion for "new stuff" and R&D.
The UK has a lot fewer soldiers, MBTs, IFVs and APCs compared to Russia and therefore it is quite possible for the UK to almost match Russia in this submarine aspect.
The UK has already more modern destroyers than Russia; UK har six Type 45 destroyers, whilst Russia is unfortunately still struggling with putting Admiral Gorskhov frigate/destroyer into active service.
The same goes for the Admiral Grigorovich frigate that still hasn't been put into active service.