Can I keep mulled wine on the stove for longer

Yes, you absolutely can do it. This is how it is done at Christmas parties and commercial Christmas markets in Germany. Just keep it warm and it will be OK. You won't get much alcohol or taste loss.

I don't think you will get any off taste, but it will change the nature of your drink - at the minimum, it will be much less alcoholic afterwards, likely far more concentrated, and possibly more spiced (depending on whether straining the spices out is something you would think to do before this long simmer)

Looking at your recipe, it looks like a mulled wine flavored syrup added to wine - the actual quantity of wine is supposed to only be heated briefly to prevent the alcohol from evaporating out. The recipe actually mentions this as a reason why the spices are infused into a syrup first, instead of the whole quantity of wine - the boiling would evaporate quite a bit of alcohol out of the wine, I think about 25% would be lost if you simmered for an hour.

If you still wanted to keep your mulled wine hot for an hour or two, I would suggest you look for mulled wine recipes that do have extended heating of the wine (maybeso something like this, though it's just the first I saw), the better to balance the spices so they don't over-infuse on you on one hand, or have the wine over-concentrate them. If you want your mulled wine to still be strongly alcoholic, you might add some spirits at the end, just before serving to balance the loss.

So, yeah, the summery is, the wine will likely not be spoiled, or develop off flavors, just by being heated for extended periods of time, this happens in recipes, including mulled wine recipes, often enough for serious problems to be unlikely - though it will change your recipe's effects. The loss of alcohol is a factor, but it can be compensated for. The level of the spices might be tricky to manage - you might want less of you're going to be simmering them in the wine the whole time, you might get less out of them (ie, need more) if you only simmered them in the wine and skipped the hard boiling syrup stage (it might balance out?), or you might simply strain them out of the syrup before adding the rest of the wine.

Of course, you might find it easier to prepare the syrup in advance, and add to the wine and heat them both just before serving - this would make the recipe quick to assemble and sidestep some of the complexities that simmering that long might cause. One of the comments on the recipe even suggested this might add a touch of complexity to the drink, since the flavors would have a bit more time to meld together.

You want to keep it as low as possible -- definitely not boiling. I use a slow cooker on low, it's one of very few things you can keep taking the lid off when using a slow cooker (or even leave the lid off).

It's possible your stove doesn't go low enough. A candle-heated food warmer (arbitrary amazon link) is another option if you have or can securely improvise such a thing. From experience, one candle isn't enough without a lid, and a very thin-based pan isn't recommended for such a localised heat source. This approach also has the advantage that you can keep the mulled wine where you want to serve it, which may not be the kitchen

Do you have a Thermos flask (vacuum flask, to be non-brand specific), or several? Clean them well (no lingering coffee scent, say) and preheat them with boiling water for 5 minutes or so. Dump out the hot water, pour in the mulled wine, wait for your guests to arrive.