I've had this problem when out of the UK. I found that MMS messages won't send if you don't have a mobile internet connection of at least H.
So if you phone is showing mobile data connection of E or G then it won't send.
H or H+ or 4G is needed to send MMS successfully.
They also won't send over Wifi.
Try swapping with mobile network you are connected to. This might improve the mobile data connection speed.