First, we need to determine the painting rate of the team. If 20 members can paint a 1500 square foot wall in 30 minutes, we can find their collective rate in square feet per minute.
The total area painted by 20 members in 30 minutes is:
\[ 1500 \text{ square feet} \]
The rate of the 20 members together is:
\[ \text{Rate of 20 members} = \frac{1500 \text{ square feet}}{30 \text{ minutes}} = 50 \text{ square feet per minute} \]
Next, we can find the rate of one member by dividing the collective rate by the number of members:
\[ \text{Rate of 1 member} = \frac{50 \text{ square feet per minute}}{20} = 2.5 \text{ square feet per minute} \]
We need to find the rate for 15 members:
\[ \text{Rate of 15 members} = 15 \times 2.5 \text{ square feet per minute} = 37.5 \text{ square feet per minute} \]
Now, we want to find out how long it will take these 15 members to paint a 1200 square foot wall. We can calculate the time required by dividing the area to be painted by the rate of 15 members:
\[ \text{Time} = \frac{\text{Area}}{\text{Rate}} = \frac{1200 \text{ square feet}}{37.5 \text{ square feet per minute}} \]
Calculating the time:
\[ \text{Time} = \frac{1200}{37.5} = 32 \text{ minutes} \]
Thus, the time it would take for 15 members to paint a 1200 square foot wall is
\[ \boxed{32} \text{ minutes} \]