[Grhm]

I have often experienced the same problem.
Modern browsers almost invariably seem to think they know better than me which version of a web-site I want, which I find infuriating.
The only way I know of reliably opening the desktop version of Fastmail on my Android phone, is to use an open-source browser called 'Zirco', which is available through F-droid.
The security-conscious will doubtless be aghast at this suggestion, as Zirco has not been updated since 2012, but, as I say, I know of no other way.

Update 16 Dec 2018:
Since the latest changes, Zirco now defaults to the mobile interface, so this tip no longer works... however a much more satisfactory solution is given below.