There are 20 slots of 15 minutes each from 9:00 AM to 2:00 PM. There are ten counters in the VPO.
From the given information and condition (1), Total number of applications scheduled = 10 × 20 = 200
From condition (3), Number of US applications = 100. Number of application for rest 3 categories = 100.
From condition (2), Number of US applications for each slot = 100/20 = 5. Similarly, number of applications of rest 3 categories for each slot = 100/20 = 5.
The processing time for an application is the same within each category. But it may vary across the categories.
Depending on the number of applications in a category and time required to process an application for that category,
it is possible that an applicant for a slot may be processed later.
The information can be shown in the table below. 

Step 2:

On a particular day, Ira, Vijay and Nandini were scheduled for Schengen visa processing in that order. They had a 9:15
AM slot but entered the VPO at 9:20 AM i.e., they had delayed by 5 minutes.
So at least 3 people were scheduled for each slot for a Schengen visa.
Mahira and Osman were scheduled in the 9:30 AM slot on that day for visa processing in the Others category

So at least 2 people were scheduled for each slot for an Others category visa.
Total applications on that day = 200
US applications = 100
So there are 5 applications of US out of total 10 in one time slot.
There were 3 applications of Schengen and 2 applications of Others in 1 time slot.
Hence, there are 0 applications of UK in one time slot.