What it entails is really up to you, but first of all it should always be noted that being a noble's mistress in a quasi-historical setting doesn't have the same meaning we put on the word today. Historically it was very very common for nobles to have consorts and lovers outside their marriage, specially when marriages were arranged as strictly political alliances. So a "mistress" Warden is not a dirty little ho that scumbag two-timing Alistair keeps on the downlow from a loving wife, she's Alistair's one true love away from a wife he detests and only married because politics anyway.
That said, what actually happens after the game is over is entirely up to your headcanon. I can't really imagine Alistair ruling alone indefinitely - he would have to find himself a wife eventually, even if that's not Anora, because his lineage is in dire straits enough as it is. Brandel only had one child, Moira only had one chid, Maric only had one legitimate child, Cailan had no children at all (if he ever had any bastards, something tells me Loghain took care of those at the height of his madness, around the same time he figured that selling elves into slavery was a fine way to raise more coin for the army). Alistair is walking on thin ice and can only take this "sion of Calenhad" thing so far; he needs those heirs, or at least the appearance that he's trying to make some. It'd be hard for him to stay a bachelor forever without the bannorn coming down hot on his neck demanding satisfaction.
As for a possible bastard with the Warden, I think it would depend on the Warden. A bastard with a Cousland I can easily see being elevated to the throne in the lack of another heir, because the child would still be noble-blooded enough, coming from two of the greatest houses of Ferelden, even if outside a formal marriage. A bastard with an elven or mage Warden would have a real hard time finding acceptance, though (I'm not even counting dwarf because that's double the almost impossibility). The child would not only be a bastard, but a bastard coming from the lowest of the lowest rings of human society. It's not even a commoner, a farmer, a tavern wench, it's even lower than that. I don't really see the bannorn putting an elf-blooded or mage-blooded bastard on the throne; they'd sooner end the Theirin legacy and elect someone else. Remember that the throne of Ferelden is not necessarily hereditary, the Theirins have only managed to stay in power because they're charismatic enough to charm everyone with a wink and a smile. But the bannorn is just as comfortable putting an end to the Theirin legacy, exiling/executing Calenhad's last heir, and putting a Mac Tir on the throne if that's what looks like the best choice. There isn't really an undying loyalty to Alistair's heritage that would make just any child of his elligible for having his blood alone.
So with all that, you can pretty much headcanon what happens with a mistress Warden after the credits roll in almost any way you want. There are some things that are highly unlikely to happen if we're being faithful loremongers, but eh. It's your headcanon. Ain't hurting anybody.
For myself, I have Alistair married to Anora, with Cousland as his mistress, as the canon for my Cullen playthrough. I figured neither Alistair nor Anora will ever want to come anywhere near each other's genitals, so no heir will ever come out of there. Meanwhile, one of the lines you can give Alistair in game when he says it's almost impossible for two Wardens to conceive a child is something like "well that's not gonna keep us from trying aaaye," so I do headcanon that after a few years and many many many many attempts, Cousland does end up producing a child against all odds (with a little help from Spirit Healer and Creation Magic specialist Senior Enchanter Wynne, shhhh nobody needs to know), and that's gonna have to be Alistair's heir because his marriage never went anywhere. I don't know what will happen to our Wardens in Inquisition, but granted that my Cousland doesn't die in game, I entertain the notion of Anora eventually getting out of the picture somehow and Cousland finally agreeing to marrying Alistair shortly before her Calling, if only to legitimize their child's claim. If she dies in game before that's possible... I kinda like the idea of Alistair formally acknowleging their child as his legitimate heir anyway, and the bannorn not even caring that much because, eh, the kid's noble enough, has good political ties between the Couslands, the Guerins, the Theirins and the Grey Wardens; and it's better than another civil war anyway.