This isn't really on point, as you were asking about how soon you can start the online application, but you said some things in your post that I want to highlight...
Your signature says your first FLR(M) was granted 11 January 2018. But then you said you "need" to apply for your second FLR(M) 1 June 2020. You won't be on your first FLR(M) for 30 months until 11 July 2020, so the earliest you should apply for your second FLR(M) is 28 days before that, or 13 June 2020. Is there a reason you said you needed to apply 1 June?