I do not know what kind of work you plan on doing with that lathe. In the condition it is in, with nothing missing, it is usable with some cleaning and adjustments. I am "old school" and know that with some care, good work can be gotten out of older and somewhat worn machine tools- if the person running them knows what they are doing. My own Southbend Heavy 10" lathe has a bad ridge worn on the bed-way for a few inches at the headstock end of the bed. Despite this, I get work out on that lathe within 0.001" or better.
The lathe you show in your pictures has mostly surface rust on it. It has adjustable headstock spindle bearings (using a hooked spanner wrench on the slotted adjusting and locking nuts). It has this same system to adjust the end-play or thrust in the leadscrew/feed shaft.
The lathe is complete, and unless you are planning on doing highly accurate or very small and fine work, it should be good for most jobs. I'd clean it up by carefully scraping the rust off the machined surfaces with the edge of a single-edge razor blade, and use some steel wool and diesel fuel or similar.
Adjustments and checks: 1. Flush the headstock bearings with diesel fuel or kerosene. Then, check the radial play in the bearings using a dial indicator and a
stick of hardwood to provide leverage. Make sure to back off the spindle thrust adjustment when doing this check. If the bearings
have somewhere between 0.0015"-0.002" on the larger bearing and 0.001"-0.0015" on the smaller bearing, you should be good to go.
2. clean the dovetail sliding surfaces on the cross slide and compound (top slide), and see how tight or loose they feel when you crank
the cross slide and top slide. Adjust the gibs but do not be surprised if the cross slide and top slide feel tight at some portions
of their travel and loose at other portions of their travel. Surfaces wear unevenly depending on where the most use occurred.
I "average" the adjustment out when dealing with this kind of wear, but leave things set a little on the tighter side.
3. Flush the apron with kerosene or diesel fuel and make sure the feed clutches for cross and long power feeds, as well as the half
nuts work properly and freely.
4. Find the lubrication point on the headstock cone pulley. This is often a set screw that is in the center of one of the steps of the
pulley. It puts oil into the bearing between the hub of the headstock cone pulley and the spindle when running with the back gears
engaged. Shoot a little automotive brake cleaner in this hole, then some kerosene or diesel fuel. You want to make sure the oil
can get where it needs to go.
5. The feed/threading reverse gearing (tumbler gearing) is inside the headstock casting on this lathe, so getting to it will be a little
more difficult.
Inspect for damaged teeth, clean and lubricate. You may find the reverse gears are a bit loose on their pins. For the immediate,
it should be OK if no other damage is found, but in time, you may want to bore/bush these gears and make new pins.
You do not say whether you got the set of change gears with the lathe. If you intend to cut threads, you will need this set of change gears, as well as to get
different feed rates. The threading chart is missing from the lathe, and this would tell you what combination of gears is needed for different thread pitches.
Without the change gears, this lathe becomes a lot more limited in its use. OK for turning and boring, but not able to cut screw threads except for the one
which the gearing is on the lathe for. You will also need to get the pitch of the lead screw and work the math to determine what combinations of gearing cut what pitches of threads if you intend to cut threads.
A drive for the lathe is not so much of a problem. Even without the matching cone pulley for a countershaft, a drive can be built. Some pillow block bearings (Plummer blocks in British terminology, I think), and some angle iron and a motor and you can buyild a good drive for the lathe. Some people have successfully made the step-cone pulleys from wood, turning them in-place on the countershafts. Or, you can make one single flat belt pulley about the size of the step on the lathe's headstock pulley. A combination of step-cone vee belt pulleys (one on the motor, one on the countershaft) will then give you your speed changes. I tend to go simple or old school on this kind of drive. For an older lathe which may not have all the change gears, it does not pay to invest in a VFD, in my opinion.
A simple countershaft for the least amount of work and money is my suggestion. Get the lathe running and put it to use.
The lathe has a threaded spindle nose, so unless you have a four jaw chuck as well as a faceplate, the work you can do on the lathe is also limited. Unless you can get chuck plates machined and threaded to fit the spindle nose, you are really limited by that 3 jaw chuck. If I had one chuck to choose for my lathes, it would be the four jaw "independent" chuck. I can adjust the jaws to chuck odd shaped work and I can adjust the jaws so any job runs dead true. A three jaw chuck, particularly an older one, is really an inaccurate and limited proposition. The chuck shown on the lathe has solid jaws, so to chuck larger work you have to have the "reverse jaws" for that particular chuck. The jaws are machined by the chuck manufacturer to mate up to that chuck. If they are lost, no way to replace them as a rule.
Other unknowns are the type of tapers in the headstock and tailstock spindles. This is important for using centers and for using a drill chuck on an arbor in the tailstock. There are no guarantees as to what taper the lathe's manufacturer used- most likely Morse, but you never know. There are a lot of unknowns with this old lathe, nothing that would stop the lathe from seeing some limited use. You seem to have the lathe in your garage, so you already own it. I'd say set it up, clean and adjust it and start using it. In the worst case, it is not up to what you need and you can perhaps re-sell it as a lathe in better condition than you got it and move on up to some other lathe. Good luck !