It stands for We Are The Opposition Now, hence the acronym WATON.
Although the concept that we, some version of the public, are the real opposition has been around as a political concept in various context for a while, its use by pro-Europeans on Twitter only started getting going at the end of 2017 and in early 2018.
In both its long form, #WeAreTheOppositionNow, and the now more popular shorter and more obscure #WATON, it’s a reaction to the failure of pro-Remain Conservative and Labour MPs to follow through with opposition to Brexit in Parliamentary votes.
Hence the idea that we, the grassroots Remainers, are the opposition now.
P.S. The very first time #WATON was used on Twitter, back in 2009, was a bit of a red herring as far as its current usage is concerned.