Rebuild chimney breast instead of RSJ

Yes you can put the stack back again using bricks (if exposed brickwork is required) Alternatively using blocks if you intend to plaster it afterwards. my guess is though the previous builder who removed the lower part of the stack must have supported it somehow. perhaps there is an RSJ running between the joists so its not visible. anyhow if its not supported then its definitely something you would want to get checked

If you would like to maintain the floorspace on the ground floor instead of building up bricks to meet stack on the 1st floor, then I would advise adopting a 'gallows bracket'. A gallows bracket is a series of vertical triangular pieces of steel (similar to set squares) connected by a horizontal metal plate. This can be attached to the ground floor wall and support the chimney stack from below. Search 'gallows brackets' in Google for more info. You may need a Structural Engineer to design it, alternatively the fabricators may offer this service and guarantee the bracket's performance. Please ensure that you employ a qualified contractor to install it with suitable professional liability insurance. Good luck!