[Arran]

I doubt whether there is anything wrong with the speaker, the 1000 ohms sited on the schematic may mean inductance rather then DC resistance, even if it is DC resistance 810 ohms is well withing the margin of error of those days of + or - 20%, if the field doesn't get smoking hot I would leave it alone. If if lets say the field winding was shorted I would take the route of rewinding the field coil before dumping the speaker, especially if you can unbolt the mounting bracket.
The output transformer isn't a problem if it needs replacement, it's a single ended power output stage with a 6F6 so that's fairly typical, and you have a good range of substitutes to work with, a transformer designed for a #42, a 6V6, or even a single ended 6L6 will work with a 6F6. Typically the output transformer primary, if it is mounted on the speaker, will share one lead with the field coil, the other end of the primary should go to the plate of the 6F6. So you need a minimum of three wires in the speaker cable, sometimes they use four if the field coild and the output transformer don't share a lead. If you are getting a reading of several hundred ohms DC resistance on the output transformer primary I would call it good, it will either be open or have continuity, it will not read 7000 ohms that's inductance.
Regards
Arran