Hi John,
I don't have experience of this particular lathe but many ideas are found on all similarly-aged lathes and I am a bugger for Drummonds. Obvously I did what you did and went to Tony Griffiths' site to look it up.
I would ask what sort of movement you found on the spindle? The threaded rings on the outside end of the headstock spindle are to take up end float. If it does not already have one, a ball bearing thrust washer would do the old girl a power of good and not be out of keeping for a machine that age.
Side play is taken up by the pinch bolts on the head bearings. I don't know if the spindle runs in the cast iron directly but I just stripped a Zyto lookalike at Forncett Steam Museum and that has thin bronze bushes split on one side. (If you are reading this Ramon, it is the one on the table next to the Sleaford engine, and has suffered from neglect in the very humid apmosphere).
Be careful when adjusting the pinch bolts that you do not crack the casting! If it looks like excessive travel, replace the bushes, but I don't think, from what you say, that this will be a problem. Make sure that you align the oil holes in the bushes (if you have them) with the oil holes in tha casting when putting it back together (forgive me if I appear to be trying to teach my grandmother how to suck eggs).
I shouldn't worry about the gap piece, particularly if you are going to be mostly using a chuck. The saddle should not run off the end (as you can see in one of Tony's photographs); it is only for very close working to a faceplate with carbon steel tools. A gap piece only supports the leading edge of the saddle, it does not add any strength to the bed as it only sits in the gap.
I hope that this helps.
Andy