You did not need remove it from the lathe again, if that's what you mean, unless temporarily as part of cleaning the machine generally.
It stays there!
In normal use you slacken its securing screw, lift it clear of the two spindle gears and clamp the screw up again. Then tighten the pulley grub-screw so the drive is now directly to the spindle.
It gives a low spindle-speed, whose primary purposes on this lathe are two-fold.
1 – It allows slow rotation when turning something of large diameter, and for interrupted cuts (e.g. turning square and hexagon bar) as that imparts a hammering action to the machine.
2 – Screw-cutting, by driving the lead-screw via the appropriate change-wheels to give the correct pitch.
On more sophisticated lathes it adds a third, a fine self-acting feed; but the EW is not designed to give this.
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The chucks and face-plate on the EW (and many other smaller lathes including the Myfords) are screwed onto the spindle nose; and they can become very tight to unscrew. Heavy interrupted cuts are among the most likely causes of excessive tightness.
The safest way here is to use two strap-wrenches (not chain-wrenches!); one threaded round the pulley, and the other round the chuck. Sometimes, simply gripping the belt tightly by hand only, round the largest pulley, is enough.
DO NOT use the horribly tempting way that too many lathe owners have done to their cost, of using the back-gear as a lock, and the chuck-key as a lever, often clouted with a hammer. It is more likely to break the gear teeth than slacken the chuck.
When screwing a chuck or back-plate to the nose, ensure the threads and registers on both components are scrupulously clean, and give them a smear of thin oil. Tighten it by hand only, to meet the register flange.
Note the danger of running such a lathe backwards, as very occasionally useful, is of unscrewing the chuck.
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For the rest…
For cleaning this lathe, leave the headstock castings on the bed. You can slide the tailstock off. In fact I do that sometimes when I do not need it there, to give me more room for the operation in hand.
A clue to the overall state of the machine is the appearance of the top of the bed. Unless it has been surface-ground at some point, the bed on these lathes seems to have been finished by fly-cutting. If the little semi-circles this leaves extend under the chuck, the lathe has seen quite light, careful use.
I would though expect wear on the steel lead-screw and its cast-iron nut as these are used for all feeds, both manual and for screw-cutting.
The spindle should turn freely and smoothly by hand, with the belt off, with no shake. Be very careful adjusting the pinch-screws, not to crack the thin journal walls.
I would expect the back of the spindle register to have worn a cylindrical rut in the front of the headstock. It was quite deep on mine, so I made a thin bronze thrust-washer, fitted with a very thin brass anti-rotation peg that just lodges in the adjustment-slit in the headstock. The peg I made is a weeny bit of brass studding screwed into the washer, trimmed to length, and with two flats filed on it.
From memory I think the gib adjusting screws are BSF or BSW, with slotted heads that over time become rather chewed up. I think some of our suppliers stock suitable grub-screws, but they could be made from stock silver steel bar. If so, give them a point or chamfer, and harden and temper them, so they don't become irretrievably mushroomed inside the slide.
(Some might suggest opening and re-tapping the holes to take metric grub-screws I would not except where a thread has become stripped! Apart from mixing thread-standards in one machine being bad practice, there is precious little brittle cast-iron around them for safety; nor room for the larger lock-nuts.)