Shoes for flat pedals?

I commute in Glasgow and wear Gore-tex trainers.
In fact, since moving to Glasgow I only nuy Gore-tex shoes now.
They kep me dry in a shower. If I'm on  alonger ride and/or the rain gets heavy I have overshoes and Sealskinz socks. But the convenience of Gore-tex shoes is brilliant; it means not stopping to put on overshoes or change socks if it rains unexpectedly andf they keep me dry enough if the roads are wet/flooed but it's not actually raining.

I keep looking at Five Tens but none of them seem to be properly waterproof. I find it incredible that there are no Gore-tex MTB shoes.

Astro turf soccer shoes

As a BMXer, I generally wear Vans. They're inexpensive and comfortable and the soles cope well with the abrasion from pedals. Water resistant they are not.

I wear Lonsdale boxing trainers (Fulham) for my 10 mile commute. Dead flat sole, plenty of grip, quite stiff, leather uppers, velcro fastening, comfy. They're waterproof for my ride but no shoe is 100% waterproof - depends on what you throw at 'em. Best bit? £25 or less.  

Have you considered goretex hill walking shoes? Loads and loads at under £100. Most have a Vibram sole, which I would think would stand up to the pedal pins reasonably well.

I use them for, well, walking in conditions like you state and my feet stay perfectly dry. Worth considering.

PP

I use 5 10 Impact Pro. They are more weatherproof than a standard 510. Claims of ‘quick drying’ are relative though - they have plenty of padding, so still need newspaper to dry them out. They come up in sales or eBay in your price range. Best to try some on, as the calibration on sizing is on the small side.

Waterproof socks any help here?

Dunlop Volleys
Heaps of grip.
Rubber is a plant.
Cotton is a plant.

Don't know about wet. I think my socks get wet in storm but I can't remember cold feet. (Keep pedaling). Where as with Shimano cleat shoes I only get a wet spot on the souls (probably cleat hole).

Dunlop Volley are cheap too enabling your budget to thicken your wallet instead or maybe not as long lasting but looks like Earths made materials. They'll grip, especially on triple traps or similar but everything else does too.

Hence my bets on Volleys

I used worn our skate style shoes for many years (Vans mostly) then pushed the boat out and got some Five Tens, the grip on the pedals is stunning.  I went for the Spitfire as it wasn't quite as ugly as most of them, but fabric front so lets in moisture.  They have lasted many years and are gradually falling to bits but still work.  Five Tens really work.

I could do with a new pair and am tempted by the Ion Raid Select, horrible front bumper, but full tan leather.