EMF measured while verification of Nernst equation keeps on increasing
You should arrive to a plateau value, the EMF can't increase perpetually. The ideal potential difference is around 1.1 V given the experimental arrange, this is also known as the Daniell Cell. I think the low initial value you get is because of the time response of the system, until it reaches a steady state.
There are basically two types of connections:

i) Electrode/Solution: Some specie B transfers an electron from the solution to electrode surface (by receiving or by giving). The electron transfer rate is very high, specially on batteries, so this can not limit the whole process.

ii) Ion migration through solution: This step is the limiting process. The ions are the charge carriers in bulk solution, which move much slower compared with electrons in a conductive metal. A way to overcome this is using strong acid solutions, as sulfuric acid in the Lead-Acid Battery used on cars.

The response you have will depend on the time of measure (immediately after closing the connection, 5 seconds after, etc.) and the time interval of such measure. It will depend also on the initial concentrations of the solutions (high or low). If the concentration is low, the ionic strength of solutions is weak and few ions are to carry the charge; adding table salt (NaCl) is a possible solution. The ionic bridge may be highly conductive but, if the water permeation is also limited, you have an extra limitation there, which will be higher as the phase thickness increases (ie using too much agar, or long bridge). A solution may be, using an U-Tube with NaCl 4 M and covered in the ends with commercial water-purifier filters well sealed to the glass.

A delay on response could be due to the hydration time of the agar gel, which can take a while. Once water has permeated the agar, the ions begin to move faster, until steady state is reached.