The rule in (i) can be modified as follows: ants meeting head-on in the north-south direction will turn clockwise by $90^\circ$, while ants meeting head-on in the east-west direction will turn counterclockwise by $90^\circ$. Comparing before and after this modification, the positions of all ants at any given time do not change; only the roles of the ants that previously met in the east-west direction are swapped between each other, so this does not affect the time at which doomsday occurs. After the modification, all ants are divided into two classes: (NE class) always moving east or north; (SW class) always moving west or south.

Let coordinate $(0,0)$ represent the SW corner of the table, and coordinate $(m, m)$ represent the NE corner of the table. At time $t$,
the region $\{(x, y) \mid x + y \le 1 + t\}$ has no (NE class) ants
the region $\{(x, y) \mid x + y \ge 2m - 1 - t\}$ has no (SW class) ants

So at $t = m - 1$, this is the last time at which ants can meet, and they can only meet on the line $x + y = m$; after this, all ants only move forward. On the line $x + y = m$, ants moving respectively toward the east, south, west, and north directions will, at most, take $m/2$ units of time to reach the edge of the table; so altogether we obtain $3m/2 - 1$ as an upper bound for doomsday.

Doomsday can indeed occur at time $3m/2 - 1$: ant A is located at $(1/2, 1/2)$ moving north, ant B is located at $(1/2, m-1/2)$ moving south. At time $(m-1)/2$ they meet at coordinate $(1/2, m/2)$, after which A continues moving east for $m-1/2$ units of time, arriving at the edge of the table. At time $3m/2 - 1$ the ants' doomsday occurs!