As to whether or not the Attacking Army could use Forges and Logistics buildings to reinforce/resupply their army?
No, definitely not. Here's why.
In history, when a Castle-City came under siege, Forge's and the like didn't survive beyond the enemy force showing up outside the wall. They were either moved to a more centrally defensible location, or the Forge itself was destroyed. (Read: Not the entire smithy building, just the Forging Equipment.) This wouldn't be hard to reflect in the math of the game either. A Forge being moved should obviously have a lower production value than a fully equipped Smithy, a far lower one. A destroyed Forge would be handled by the Smithy being unusable for X turns until the Forge is repaired/replaced. Basically, that means that for X turns, all Military units built in that city would lose access to whatever equipment the relevant building was giving them access to. (I.E., Smithy's, Fletcheries, Bowyers, etc etc.)
Multi-Stage sieges would be awesome, and they wouldn't be as impossible to handle as you're all assuming methinks. Multi-turn or not, even Turn-Based Tactical Battles lend themselves well to this style of combat. More-so than RTS games actually, since it's much easier to save the locations of units.
It could be handled, as I mentioned, either over the course of multiple turns, or even in a single battle.
If a multiple turn style was implemented it would be dependent on the size of the city first. If there are no walls to break down, then the only real stage should be the keep, since attackers could flood in from any number of directions on the city, making defending it unfeasible for the defenders. If there were walls, but it was a small city, then the walls would be the first stage, and the Keep/Palace the second and final stage. In larger cities the first turn would be dedicated to trying to take the walls and the City methinks. Urban combat wasn't really a big thing until the advent of Black Powder weaponry, since combat would be inefficient for both sides in such a tight space and whilst also having to deal with melee units blocking your firing zones. For there to be a third stage, the second stage would have to be an inner wall defending the Keep, with the third and final stage being the actual Keep. This would increase the need for Siege Equipment as well. Catapults and the like are obviously needed to take the outer wall, but unless you create a massive hole in the wall, or batter down the door, your Siege equipment would be stuck outside the Outer Wall... This wouldn't necessarily mean they couldn't hit the Inner Wall, it would just make them a lot less accurate methinks. This would make Battering Rams serve a thrice important purpose. Breaking the Outer Perimeter, Allowing Ranged Siege inside, and Breaking the Inner Perimeter. Catapults would become more strategically relevant, since their effectiveness could potentially be reduced before reaching the second stage of combat. This would also DRASTICALLY increase the importance of Siege Ladders, particularly in taking the Inner Wall, where your siege equipment might not reach.
At the same time, all this makes Siege Equipment the extremely valuable resource they should be. It's not like Catapults, Trebuchet's, and Siege Towers are all easy to build or anything. (I have actually built a Trebuchet before, so yes, I do have a frame of reference for saying that. They're sick btw. And no, -NOT- a full size Trebuchet, that really would be a nightmare... More like 1/4 or 1/3 size.)
Multi-Turn would be the best way to handle it though, because of the inevitability of retreat. Allowing you to recall troops to the Inner Wall/Keep when you're losing, thus sacrificing the Outer Wall, and by consequence, the City, adds a level of strategy I've not really seen in... Well... Any game I can think of. Stronghold did it half-way decently, and Lords of the Realm did it in an alright fashion too, but I wouldn't say either of those really captured the essence of what a Siege should be like.
A Single-Turn, Multi-Stage fight would be handled almost exactly the same, just remove the over-land turns in between. Outer Wall would be entirely the same, but after that, the pace would change for a few Tactical-Turns at least. Provided you still had the option to retreat to the Inner Wall, (Which would be a must in either method if you ask me,) once you did, the enemy would then traverse a number of City Tiles trying to reach your Inner Wall. This would simulate the time it takes to reach the Inner Wall, during which they take fire from the fortifications there. After that, it would continue to follow the model above again, with the battle for the Keep. (Or, more accurately, the Keep Grounds, unless we actually would get to fight -inside- the Keep, which would just be plain sick.)
The only real difference in these implementations would be Realism and Pacing. Multi-Turn would sacrifice some Pacing for Realism, while Single-Turn would sacrifice Realism for what I assume would be better game Pacing.
The other issue that would need to be addressed is how to handle retreat in Single-Overland-Turn implementation. I would prefer it be handled very fluidly, even in Turn-Based. Basically, you press the retreat button, and all surviving defenders retreat to the Inner Wall automatically, while all surviving attackers, barring Siege, advance to just inside the Outer Wall immediately. We bar Siege because of dealing with variables. That's not to say it wouldn't advance though, they would just be right outside the main gate, or a hole in the wall that they've created. The variables are, if you haven't broken down the gate or cleared away rubble, the chances of your heavier Siege Equipment actually being able to make it inside the city would be very low... And honestly, that's probably a good thing. Having Siege for two out of three stages of the fight would probably be a little imbalanced actually... So you might just 'abandon' Heavy Siege Equipment at that point, opting for nothing but Ladders and Battering Rams. ('Abandon' : Unavailable for the remainder of the Tactical Battle.) The abandonment wouldn't be hard to implement either, either just disable the Siege equipment, or make it impossible for them to get inside, and always have them be out of range of the Inner Wall. Siege Spells, however, should still work just fine, like Ladders and Rams. Actually, it would be imbalanced, and would completely defeat the purpose of retreating to the Inner Wall, so yes, no more Siege after the first stage.