Labor here in Australia have won in the past with little more than 40% of the primary vote...so not that different from UK Labour with their 36%.
It is interesting to note that 70% of the public voted for parties that were and still are pro the Iraq war - Torries and Labour. It seems the impression conveyed in the press until recently of vehement majority opposition to the war was a misrepresentation (lie). Also, immigration played a much bigger role than many in the press are willing to admit.