Hello Aimee,
It's always an added pleasure for me to assist a fellow Guest from the United Kingdom with their jolly holiday plans to visit Walt Disney World Resort!
When my own Mouseketeers were little we always used to hire a
Rental Stroller from Disney during our magical stays and of course it always made financial sense for us to utilise a pre-paid
length of stay rental rather than just to hire day-by-day.
Of course another great option for you to consider would be to use one of the
Disney Featured Providers who can arrange both a hassle-free delivery and pick-up direct from your Walt Disney World Resort hotel.
In addition to all this rental information I highly recommend that you double check with your airline what their current carriage rules and regulations are for small collapsable, umbrella style buggies or strollers is - as you might just find that they don't actually incur any extra charges at all!
I also thought you may find this
Baby Care Centre link to be a useful planning resource for your dear family.
Wishing you an extra magical holiday to Walt Disney World Resort where I know you'll create some unforgettable memories which are sure to last an entire lifetime.
Have a great Disney day.
~ Jane