I believe that a centrist party in a system that isn't rigged (like our one is) would have a massive following, as would a lot of smaller party's but I don't believe that the bulk of hard-core Labour or Conservative voters would ever vote for them.
Trying to have centrist leaders and MP's in the Labour and Conservative party's and appeal to core voters is like a trying to put a square peg in a round hole.
The problem with this theory - and the reason why it has been found to be false - is that the electorate and the political class have entirely different understandings of what "centrism" is.
The public thinks it means "not extreme", average, middle-of-the-road sort of politics, which is why claims that a politician is extreme (one way or the other), radical, fights against the orthodox (of whatever side) etc are so often deployed by opponents and the papers to put people off voting for them. They think it means politicians who believe in what the majority want, even though that is something thats impossible to define, is very vulnerable to sudden events / manipulation and which shifts over time anyway. People are also very reluctant to ever admit that they were wrong, which is why something that was centrist once (like the hostile environment, which both major parties agreed with) is now no longer centrist.
Politicians on the other hand believe that it is a definable type of political belief - the same sort of actually quite right-wing, pro-market, pro-EU, pretending to be a bit paternalist careerist nonsense that people are almost required to adopt as a faith before they get on in any of the main parties.
When the public are confronted with actual centrists standing on an actual centrist platform, they (at least given the evidence of the past twenty years) usually run away; when only centrism is on the menu (as it was in 2010 and 2015), they either hold their noses and vote or do not vote at all. It is only when centrist parties masquerade as something else that they get success.