I thought at the end of men at arms Carrot held on to the evidence collected by Edward and put it in a safe place, Vetinari knew this.
Vetinari took Carrot to inspect the old throne and talked about politics, Carrot indicting that there may one day be the need for a king.
I got the impression from the book that Terry left the door open for the return of a king.
This is how I would set Carrot on the throne......
First I would have Vetinari either become so ill He had to step down or die and thus a power play for the patricanship starts with the guilds fighting for the right to rule the city.
This leads to civil war and Vimes is approached and asked to take up the post of Patrician but refuses on two grounds. Firstly He dose not agree with the way the patricanship is so powerful and is open to abuse and secondly He fears for the safety of young Sam and Sybil, they could be used as pawns in a power struggle .
Vimes approaches Carrot, who just about everyone in the city knows is the rightful King, and asks Him to take up the throne and reunite the city states....... but carrot is not agreeable to the idea.
At the same time Klatch and Überwald are both watching the city falling apart and start to amass armies on their boarders to annex the city as neither of the two can afford to allow such a prize to fall into the others hands.
Vimes and some of the watch infiltrate these countries and gather evidence of the impending assault.
They return and present the evidence to Carrot who they convince to take up the Kingship to reunite the city under one banner.
Vimes wants some controls in place as insists that a parliament be set up to counter any Royal excesses even though He trusts Carrot to rule wisely.
Carrot is crowned, somewhat reluctantly, but klatch has already made a move to invade, so carrot marries Angua who is as near to being a royalty that Überwald has and thus seals a bond between Überwald and the city.
This means that if Klatch go ahead and overrun the country they will put them self in a position where they have insulted Überwald and put them at war, so klatch retreats back to its boarders and Überwald and klatch withdraw their forces and the status quo is returned.
Then Vetinari either dies and comes back as a zombie or it turns out He was a vampire all along, He is offered the place in parliament as permanent Prime Minister and accepts the post.
The city is now run on a two tire system, Royal blood and selected ministers from the guilds.
There is no voting for the masses, the ministers are elected in house by the guilds.
A follow on book could have its basis set around carrot and angua raising a family.......
One of who is a Male werewolf and next in line to the throne, one a Female Human who is Third in line to the throne and finally and one a Dog who is Second in line to the Throne.
You could have all sorts of fun and infighting politically.
Will the city accept a Werewolf King, if not how will Überwald react to such a situation... or would a Dog ever be accepted as King, if so the I see a post for Gaspode as interpreter to the King....... or will the crown go to the Female line even though there are Male heirs alive and kicking..... you could do so much with a situation like this.
How do Carrot and Angua cope raising a mixed family, how do the young ones get on at school.......... and so on.
Or maybe Carrot is not the next King and was just a child belonging to one of the servants from the wagon train that was ambushed and the line died out at Copperhead.......!