Ok Roy
Without all the details it will be a bit of a general idea.
First support the upstairs wall, probably with needles as you're fitting between joists, the number and positioning of these depends on the wall. As it's now a wooden beam I would guess it's old brickwork, lime mortar, easier for cutting out for needles but it dosen't stay together as well. Plus you've said there's already some movement in the wall so for safety this stage has to be done right.
Once the wall is supported, put acrows under the existing beam, they are only there for support, not overtightened, that will lift the beam.
Next cut out for padstones, checking the existing brickwork. It might need remedial work or a new piers. I would probably use pre-cast padstones as they will allow more room for fitting the steel. They can be fitted after the steel is in place.
Next take out old beam and lift new steel into place.There are a lot of options on how this is done, it depends on the site. It's down to experience on the best way. Usually I'll do it manually with acrows or staging or just muscle, you can use lifting dollies but these I would keep for heavy steels.
We'll say that no bricks have fallen out, keeping it simple, we find the lowest point on the brickwork and put a 15mm wood packer on the steel at this point and jack the steel up level. You only jack the steel on the ends, never in the middle, and again not overtightening the acrows.
Bed the padstones on leaving 15mm between the top of the padstone and the steel, this is then packed with a semi dry mix, I use one part sharp sand to one part cement. If there's been a bit of work done under the padstone then this packing might be done the next day to allow for any shrinkage in the mortar. Then brick steel in on top of padstone.
We now pack between the top of the steel and the brickwork. I use a semi dry mix, three sharp to one cement. I don't mix it dry, I mix it wet and use it when it's drying out. I do it this way so there's less shrinkage in the mix. You can use an addetive called, I think, combex, I've used it but didn't find it any better. You find some builders pack the steel with bits of slate or brick, this is not the best way, it's just pressure points, which can alter with movment in the wall. I've seen lots of slate packing where the slate is easily pulled out, there's been movement and the pressure has altered. The best way is to pack it 100%, no gaps, and a cement mix will do this.
The 15mm wood packer is the tightest point on the steel, it's the minimum that you can easily fully pack. With a wooden beam there's normally a sag in the centre which will give a bigger gap towards the sides, if the brickwork was quite level I'd use a bigger packer, you need to make sure you fully pack this gap easily.
Leave the packer in place then get the mix in using trowels, fingers, whatever, and ram it tight with a small length of timber or your fingers. If you can get your hand up the back of the steel and push the mix into it tight from the front, it's easier, but it must be packed fully. Normally you would pack from below the steel but fitting between joists might mean doing it from above, which might mean floorboards up. You have to do whatever it takes to pack it properly.
When all the packing is done it's left for a few days to go off. Take out the wood packer on the steel and fill the gap. Take needles out one at a time and make good the holes, then lower the acrows on the steel slowly, you have to check there's no movement. Now make good on walls and ceiling.
There's so much more to consider, bricks that come loose, getting the steel into position, size of padstone etc. I've kept it simple as possible.
This is only to give you an idea of what's involved, I know the way in my head , I hope I've explained it enough for you.
The things you could look at doing are hacking off plaster so the builder an see what's involved, cutting holes for the needles, padstones etc. all the making good. Most builders will avoid decorating. You might help with the labouring and get something off the price.
Personally I would do it all myself, apart from maybe the decorating, that way it would be down to me if anything went wrong, and you might find most good structural builders will be the same. Sometimes you have to bite the bullet and pay the money, it's hard parting with your hard earned but there's high prices in the building game at the moment and it's attracting a lot off chancers. If any builders tell you that you don't need to pack fully between the steel and brickwork ie. use slate instead, then they will not be giving you a top notch job, for your money you want the best job, so try for that.
You now have some idea of what is involved and what to look for, hopefully you'll find a good builder at the right price.There are some really good builders out there, hope your S.E. can put you on to one.
Hope it goes well for you.