I certainly don't think the media directly encouraged people to show their emotions or were pro-actively responsible for the national outpouring of grief. Early in the morning the media were outside the palaces and within a couple of hours of Diana's death, a trickle of people started to bring flowers to the palace gates. People saw this and thought they would do the same and within a day it grew and grew into the mass of flowers.
Other than that, and the initial shock people felt, people pretty much got on with their lives, went back to work after the weekend, talked about it a lot etc. Everything remained open, cafe's, theatres, shops etc, and they were all as busy as normal. No one should have felt awkward about doing normal things - I certainly did. It was a strange week, though and our lives were saturated with the news and TV reports.
Most shops shut on the morning of the funeral, and if they didn't they would certainly have had a loss of trade for a few hours. I lived in my home city of Norwich at the time and after about 1pm-2pm the shops in the city centre were open again and everything seemed back to normal.
What the media did do that week was to whip up a storm against the Queen, which I think many people did take notice of. Headlines such as "where is the Queen, where is her flag" and "the nation needs you ma'am" - or some such thing - really caused a problem. I really don't think most people would have thought or worried that the Queen and her family were still at Balmoral had it not been for the media.
ETA - for my part I was very frustrated at having to go back to work because I just wanted to stay at home and watch the news. I had spent all the Sunday trying to take in the news and changing channels all the time to see as much coverage as I could. There was definitely a quiet atmosphere in the office on that Monday morning.