It depends what the wall is made of. Fifty odd years ago, they use to make a skeleton construction of steel girders and then add a honeycombed wall structure, usually with lots as asbestos. You may be lucky and just have a "normal" wall.
Either way, you need advice, your LA will have a team that can come and let you know what to do. It may be very easy, the steel girders in our old build used to let you just cut a few beams out and patch up, very easy and we did it a few times. If its a solid wall, then it might just need a "slot" cutting so you can fit a new beam, then take the wall down below it.
There may be another option, think outside the box, literally! Are there any other rooms that you could use instead of these two? narrow think rooms make good store rooms/server rooms and in the longer term this might be a better idea.
Phil