Packing just one backpack between them, the family packed up their belongings, sold their car and put their London house up for rent - and Mr Bayes left his job as a landscaper.
Ms Edwards, a nurse who grew up in Ireland, said her family thought she "was completely nuts" when she revealed their plans.
The 31-year-old said she stored the family's liquids and lotions in empty baby food pouches to make extra room in the backpack.
Since they first set off, the family have visited Ireland, Singapore, Australia, Indonesia, Malaysia, Vietnam, Taiwan, and even Hong Kong - with Ms Edwards breastfeeding Esmé, who was given travel vaccinations before they left the UK and in New Zealand.
Ms Edwards said they were able to fund the trip - which she estimates cost about £10,000 (NZ$22,000) - through her maternity pay with the NHS, and Mr Bayes doing landscaping work in New Zealand to "earn a bit of extra cash".
Ms Edwards also said that they booked most of the flights months before departing the UK, while they were both earning a full wage. She added that the trip was made more feasible by the fact that she uses air miles and was given a BA companion pass, which saved them £2,000 (NZ$4370).
After returning to London and spending a short time at home - which Ms Edwards says has been the most challenging bit so far as they're having to find ways to entertain Esmé, who is now 17-months-old - they're set to jet off again next week to Thailand, Cambodia and New Zealand.
Speaking to MailOnline, Ms Edwards, who first took Esmé travelling in November 2014 and returned in September the following year, said: "I was thinking about maternity leave and decided I'd like to take her travelling.
"Both Shaun and I were keen backpackers before I fell pregnant and everyone thought we were completely nuts when we said that we were going to take Esmé when she arrived.
"She was really well behaved and because she was small it was easy enough to keep her in the harness when we were on the move.
"We have no regrets, she got to experience new sights, scenes and smells as she grew and that is incredible.
"The only time I felt Esmé was missing out was not having much socialising with other children.
"However the libraries always have free rhyme classes if you need to socialise your child a little.
"The weirdest part of the trip was actually coming home. Esmé hasn't lived somewhere for a long period of time because we were always moving around hotels and rooms.
'It's also harder to find her things to do in London - well, compared to travelling anyway.
"People say having a child ruins travelling, but it really doesn't. Obviously there were times, like at night in the hotel room when you fancy a beer and the baby is asleep, and when you crack it open she wakes, that makes it harder. But it's worth it.
"You can still do everything, including snorkelling and diving, but you just take it in turns. I can't imagine having spent my maternity leave any other way."
Ms Edwards also admits that travelling with a baby is sometimes hard. The main issues include the family all being in one room together when Esmé needs to sleep - as well as having lots of people around them while travelling, and while their daughter is having a tantrum.
She said that not having enough toys around can be a problem, and not being able to enjoy dinner - hurrying in a bid to "prevent cranky time".
Ms Edwards has set up a blog about her travels called Travel Mad Mum and also posts photos on her Instagram account. On her blog she wrote: "As relatively new parents, we have already been on many travel expeditions with our baby: several road trips, long ferry journeys, long and short haul flights, stopovers, camping trips and much more planned for the coming months.
'It is through these experiences, I want to share factual information, personal stories as well as top tips for travelling with little ones.
"The overall aim is to inspire like-minded travel-loving parents - 'your travel days do not have to be over'. It is all about incorporating your little person into your way of life."
Karen Edwards' top tips on travelling around the world with a baby
1. Plan ahead - Check out the areas you're planning to visit. Especially for things like malaria. We purposely visited countries that were low risk for that reason.
2. Pack less - Whatever you're considering taking, cut it down. There are things that we took that we could have done without. Also bring clothing like harem pants and dresses as they're comfortable and take up less space.
3. Bring a harness - You will definitely need some kind of baby-wearing contraption if you are planning on taking a young child abroad.
4. Take new small toys - Load up the iPad with apps for little ones on plane journeys.
5. Pack enough supplies - Bring lots of baby's favourite snacks. There's not a great selection in Asia in comparison to the UK. We ran out.
6. Don't overdo it with nappies - You can get them when you get there.
7. Try and breastfeed - I think breastfeeding is really important. It is not fully known if mum passes on immunity from travel vaccinations. However with all the other benefits you would think it does something.
8. Don't miss out on experiences - You can take turns with things such as diving and snorkelling.
9. Never be scared - Let your baby try local foods to develop their palate.
- Daily Mail
