Ms Coomber, from Dean, in Somerset, had been favourite for gold in the high-risk event at Salt Lake City.
She practised for the sport - reintroduced after a 52-year absence for being too risky - using an adapted tea-tray designed by husband Eric.
But the 28-year-old Oxford graduate was edged out in a tight contest.
Back in the UK on Saturday, she confessed to blocking out the attention her exploits had received.
She said: "I knew there was a lot of interest in the sport, which was excellent, but I could not take any notice of it until afterwards because I had to stay focused.
"It has been absolutely hectic. I don't think I've had time for it to sink in."
Ms Coomber, a RAF intelligence officer, admitted the experience of skeleton bobsleighing can be frightening at first.
She said: "It is scary when you first start - if you are human you are going to be scared.
"When you get to my level you are still scared, but you are scared of making mistakes rather than of hurting yourself."
She also said she would be back for the next Olympics adding: "I will be there in 2006 and going for another medal.
"I'm still ranked number one in the world and an Olympic gold medal would finish everything off so I'll be back."
Ms Coomber said the blizzard conditions during the race had not helped the bid for gold.
But her bronze was the first Winter Olympic individual medal since Nicky Gooch won a speed skating bronze in the Lillehammer games eight years ago.
Her mum Rosemary greeted her at Heathrow airport.
She said: "I am absolutely thrilled and I am very proud of her."
Mrs Coomber could not travel to Salt Lake City to watch her daughter in action because she suffered a heart attack 14 months ago.
She said: "I could not go to see her myself because of the high altitude - I didn't want anything to go wrong that might have put Alex off.
"The last thing she would have wanted to hear was that I was ill in hospital."
Mrs Coomber said her daughter became interested in the sport when she was invited to take part while in the RAF in Germany.
She said: "Alex had a go and was very good at it and she was offered a place in the British team.
"I have mixed feelings about her taking part because it looks so dangerous.
"She has always been a very daring girl - always ready to have a go at anything."