A union is a membership organisation that looks after the interest of it's members, I don't see how they could have abandoned people that aren't members. What has happened in other Nordic countries though is that politicians have worked hard to lessen the power of unions by decoupling things like unemployment and introducing other form of employment. Therefor it makes less and less sense to be a member of a union. But that is to a large extent a political question not a choice by the unions.
I also think maybe one of the biggest unanswered question with basic income is how you can guarantee it long term. We already know that things like minimum wage, unemployment benefits, student benefits and social insurance don't keep up over time despite the power of the institutions and the unions. Now people are saying that we should put all those things in one basket and hope that the politicians don't screw us over later?