Upvote:1
I'm not sure if you're traveling United or British Airways, but neither will charge to transport a wheelchair.
For British Airways, their wheelchair information is listed online (scroll to "Assistance for mobility and wheelchairs"), including maximum dimensions. They have a special department you can call for further information if you have any questions, want to request assistance at the airport, or need to arrange accessible seating on board. You can also request these services online. It's your choice whether you'd like to check your chair at the check-in counter or keep it until the gate, and if space permits, they may have room for it in the cabin instead of the cargo hold.
United's policy is similar: "United Airlines will accept a customer's wheelchair or other assistive device in addition to the checked baggage allowance at no additional charge." They have further information online, and you can contact them or request online if you would like assistance at the airport or an aisle-chair on the plane.