Welsh Rarebit in a microwave?

When considering your options, you'll need to remember that the cheese mix must have a certain cooking time independently of the heat source:
The flour needs some time to gelatinize. Basically, the flour, butter and milk in your mix form a roux. If you heat it too short, you'll get a floury sludge instead of the cheesy creaminess you are looking for.

So while it is possible to cook the cheese sauce in the microwave, it does save neither time nor effort. You need to heat the mix slowly and stir often. If you have only a microwave at hand or are very prone to letting the cheese mix burn a bit on the stove (and detest the pot-scrubbing), it certainly is an option worth trying out. You'll need to decide for yourself, whether it fits your cooking style.

Side note re. other recipes as mentioned in other answers:
I chose to tailor my answer to your favourite recipe because even as there are lots of variations in Welsh Rarebit recipes, I assumed you like yours (with flour, sans beer or mustard) or you wouldn't call it your "staple lunch".

I'm a Welsh Rarebit Heathen. I get 99.5% of the WR experience with this simple, microwave-friendly approach. (Mind, I see that your complex approach does not have any beer, which I thought the traditional method was supposed to, as pubs generally specified the cheese and the beer that went into theirs. And I was fairly sure there should be mustard as well. Perhaps you are also a Heathen?) Wikipedia seems to allow for "wide variation" so long as it's a cheese sauce.

Anyway, I put mustard and cheese on a slice of toast or bread, pop it in the microwave until the cheese melts, possibly stir the mass on top of the toast with a knife if it's a bit uneven, and eat. Very quick and easy. I don't bother with beer, unless I happen to be using a beer-based mustard.

It likely depends on how willing you are to modify your recipe. With a roux thickened sauce, I'm not sure how well it would translate to the microwave (though you're welcome to try, following Stephie's answer)

On the other hand, you can probably look for other thickeners - the point is, you need a thick liquid to melt your cheese into, and the thicker it is, the less likely your cheese is to clump up instead of melting smoothly. You could try using corn starch, that might work better (flour taste is more obvious when it isn't cooked enough, corn starch is more subtle).

Or you can try other thickeners - I've used a hot-water cheese sauce powder, just a spoonful to thicken which doesn't add a lot of flavor, since it's mild enough to be covered with the sharp cheese and other flavorings in the sauce. Or any other hot-water sauce powders that won't compete in flavor too much, maybe a touch of bullion powder, or instant gravy, or some such thing - just a little bit to thicken the milk, but the added flavor is mild over the whole recipe's volume. I've used mustard powder, or egg yolk powder, which both thicken a bit more in powder form - my version usually uses liquid mustard and egg yolk which help thicken and emulsify even wet, but while the yolk powder makes little difference other than extra thickening, but mustard powder is usually a lot spicier and less flavorful.

Sometimes I will mix a whole egg in, and stir often - the white firms up and ends up acting like a thickener, the yolk gives flavor and helps emulsify, though it helps that I don't mind the egg or the fact this can be somewhat lumpier, it tastes fine to me. Or I might start melting the cheese with the mustard and mushroom ketchup (which I use instead of Worcestershire sauce, similar flavor profile but I prefer it), and only add the egg yolk after, and milk a bit at a time once that's reasonably melted and mixed. For very sharp cheeses, it helps to let them sit in the milk for a while first, it hydrates them a bit more and lets them melt better.