For cleaning the machine the safest is plain sewing machine oil. With a bit of effort and swirly movements (microfiber cloths or cotton balls) it will lift up dirt and grime. Oil is what they use to smooth out the surface as they finish off the shellac polishing (that's the lacquer they had back then). Shellac is used on both the machine head and the wood finish. It's not expensive, it's usually bought as flakes you dissolve in alcohol. It's a bit different that modern paints and transparent lacquers, so it's best to read up on how it's done before you jump into it.
For cleaning very dirty machines you can use microfiber cloths wringed up in mild soapy water. I start off with the mildest approach forst and then sort to the more heavy stuff as needed. As a last finish you can polish the machine with resin type car polish. If you have discolorations or stubborn grime the first cleaning is a bit more work, later you would just wipe of with a dry soft cloth, maybe now and then polish with a smooth non-gritty resin polish. Sugar soap is on the harsh side, if you need something extra try a mild natural soap or dish washer liquid first. Shellac is a strong and durable finish, but needs to be treated gently when it comes to soap and water.
For the veneer; a steam iron will make the wood more flexible again, not as brittle and will allow for it to be pressed flat back on the the base. Take a thin tea towel, or pice of light cotton fabric, lay it on top of the wood and gradually humidify the wood using heat and steam. You really don't want things too damp before you apply gule and clamps. It's a durable repair, making the cabinet stands up to regular use again.