I live in Worcestershire

You don't say which of the two pictures shows yours. However the site does go on to say that no chemical controls are available for either (though there was one for fusarium years ago). It does say that healthy grass can withstand attacks, which is not very encouraging for you as fusarium at least can spread quite quickly.

Lets hope for some drier sunny weather to get the grass growing strongly again. sorry not to be more encouraging. I hope someone else will be able to make another suggestion for you.

Fungi growth inn your lawn is considered as a sign of a healthy lawn.

What, even fusarium and snow mould? This isn't mushrooms.

Sorry Steragram, my fault, I didn't read it correctly. I saw fungi as in mushrooms/toadstools... Should have gone to SpecSavers, LOL.

Just had a quick look at the RHS web pages on lawns - confirms my thoughts that lawns are more trouble than they're worth.
Comment on GQT a couple of weeks ago was "I have grass but I can't be bothered with a lawn" (I think it was Bob Flowerdew)