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For the UK to be considered truely 'Neutral' they would have to be willing to sell weapons and/or provide support services to both sides.
The UK has not declared itself 'Neutral' but also has not declared war on Russia or committed any actions that could be considered acts of war under international law. In effect, the UK (along witht he US and NATO etal) have been operating under the Cold War era concept that as long as you do not directly attack with any your own forces (including troops/pilots on the ground/in the air, and such), or provide a base of operations for a belligerent, then you have not committed an act of war.
Hence, Russia is not supposed to attack a country providing munitions, training, or intel, to its enemy, but if the Ukrainian ships were to leave the UK armed and prepared to make war anywhere other than the Black Sea, the UK would be considered a belligerent. The only way (in theory) that the minesweepers could enter the war without causing the UK to be labeled a belligerent is if they could transit the Bosphorus Straits and enter one of Ukraine's ports first. Once they have been 'delivered' to a Ukraine port, so to speak, they could then begin operations. Since the Straits are currently closed there is no way for this to happen.