This is naive, Russian oil and gas was traded freely on international markets until... oh, an invasion happened and the West needed to scramble to try and reduce reliance on Russia and make up for that gap.
Guess what having more domestic production allows for...
Sunak and Shapps have explicitly presented this policy as increasing the UK's energy security not that of the "West" as a whole. Given that extraction of these relatively modest* quantities of new fossil fuels is still many years away they won't help us with fuel shortages any time soon. In the meantime, increasing our energy usage efficiency and accelerating our renewable/low-carbon energy generation programs would negate the need for them.
*According to the Climate Change Committee, even if we extract all proven UK gas reserves and resources from new North Sea fields that would only meet about 1% of European gas demand each year up to 2050. New oilfields would supply at most around five years of demand (if none was exported - currently 80% are exported), according to analysis of government data (from Freedom of Information requests) by the campaign group "Uplift". It also found that together,
new oil and gas fields in the North Sea would only be enough for an additional three weeks of energy a year.
Furthermore:
'...The government says increasing UK oil and gas extraction would help reduce our reliance on countries like Russia. But there are several problems with this argument. Most importantly, that even if companies extracted more oil and gas from UK land and waters, they’d be under no obligation to sell it to UK energy companies, nevermind at any lower price. The price UK consumers pay for energy depends on what’s happening in global commodities markets - not what’s happening in UK oil and gas fields.'
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To add to that, historically OPEC has controlled the global oil price by pumping less oil when the oil price is low and more oil when it's particularly high (if it is politically expedient for them). Therefore, if new North Sea oil was being extracted and sold in sufficient volumes to significantly lower the global oil price then OPEC would simply pump less oil to increase the price again.
To create a direct link between UK oil/gas extraction and UK usage, the government would need to introduce trade tariffs or other measures to make it more attractive to the license-holders to sell oil/gas extracted here into the UK energy market. However,
the government has never said this is a part of its plan, which is odd given that they say this is about maintaining energy security for the UK. It looks like the "we need to do this to maintain energy security in the face of Russian aggression" narrative is just a convenient excuse to justify doing what Sunak and his followers have wanted to do for many years but just couldn't get away with.