Hi everyone, this is more of a helpful info thing than a question because I couldn't find the answer myself before I went so wanted to provide one for anyone else looking.
Recently I flew with my 4 month old on a Ryanair flight and was able to take my Doona into the cabin with me.
You will need to book and adult seat not a child's. (we found this to be really cheeky because it costs more to give a baby his own seat than it would to give a 5 year old). Once you've booked the seat you then need to call customer services and explain that the adult seat must be changed to an EXTRA/INFANT seat. Customer service will change this for you and ensure all seats in your booking are together.
We suggest you put the Doona next to the window facing forward with the seat belt looped through the back hook.
On the flight out the cabin crew asked us to take the baby out of his Doona any time the seat belt sign was on and put him on our laps.
With the flight back, crew were happy as long as the Doona was strapped to the seat. He slept the whole flight until landing when the air pressure got his little ears.
For us it was a cost and ease thing. Ryanair charge £50 return for your infant to sit on your lap. An adult ticket for this trip was £85 return which allowed our baby his own 10kg bag allowance. It also saved us paying £75 to hire a car seat in our destination.
Hope this helps other parents who have a Doona and are worried as to whether they can take it on board.