During last year's contest I found that the price in the market had an impact on wall building. I dump enough bricks just before the date changes into the market to bring them below 100
and a quantity of 140+. IG should sell for <450 and timber <65. These values may differ this year since the trade setting is different.
The towns will only place walsl on the dates specified. The first town to place is Edinburgh. If you are fast enough, you can dump the necessary goods on the building date, but I'm too slow for it so I dump just before the changeover. The maximum the town will build on each day is either 12 wall sections or 1 gate. Gates take less bricks, I think 25, but may be wrong. So, if the wall is 150 sections and 3 gates the fastest time to get the wall placed is 128 days (13 * 8 for walls + 3 * 8 for gates). Currently wall building is going on in my hometown plus 4 other towns. At least one will be don with the 2nd wall by the time it's time to get the new production town going. But I have 80 brickyards and counting. (There was a reason for my madness of accepting low standings in 1300 - so I hope)
Another tidbid: as long as the wall is completely placed, but gates are not yet completed, sieges will not happen.
AFAIK, there is no direct way to sponsor, or speed up, tower building, short of doing a whirlwind tour and becoming mayor for a day in each town (works only for Hanse towns since you can't be elected in others). Also, a seige seems to trigger some activity for the town. Leaving a lot of bricks in the market place may cause the AI do do foolish things, such as building houses and plants where I don't want them. On the other hand, the AI cheats and builds even without bricks.