Mossgo pro is made for this.
Personally I've never used mossgo branded chems, but I have tried others with the same active ingredient & the results were disappointing. It kills the moss fairly rapidly,.. in a couple of days it looks dead, BUT the dead moss takes forever to fall off! I treated my shed roof about 5 months ago & whilst the moss is dead it hasn't washed away yet!
I'm doing a roof clean job tomorrow & I'm using sodium hypochlorite (bleach) like the americans do. I did a test patch last week & the algae went almost straight away, but the moss stuck,... so I got out my WFP pole with an old brush & gave the roof a scrub (Dormer bungalow & smooth roof tiles) and it came up ok,.. it came away much easier where it had been treated with the SH even though I allowed only 5 minutes dwell time.
I think Mossgo would be great for algae & for maintenance cleans,.. but the initial clean is going to need some manual scrubbing IMO.
I'd love to find a chem that would do the job on its own, i reckon there's a pile of money to be made!