When you rewind the field coil(s), you need to make a simple former from some bits of wood, to get it/them exactly the same overall size and shape as the original, to fits properly around the stator pole pieces. Avoid sharp edges and/or line it with PVC insulating tape to make sure the wire insulation is not scratched.
The enamelled copper wire just needs winding on the former, with the same number of turns (and wire gauge) as the original.
(The modern stuff is typically insulated with polyurethane, but it's still sold as enamelled or "magnet wire").
Put some bits of string across the former slot first and tape the ends to the wood so they do not get lost or in the way. When the coil has the correct number of turns you can release the string ends and tie around the coil so it's held in the exact shape and does not fly apart when you dismantle the coil former...
Then tape it, removing the string ties as you go - ideally with heatproof tape, not PVC insulating tape..
You could soak it in polyurethane varnish before taping, while it's still tied in shape.
You could also use plain cotton tape and saturate that in varnish as you apply it, so it sets in as part of the overall coil.
A common motor winding tape used to be called "Empire tape", which is not made now, though it's still a well known name - this is an equivalent:
https://www.ppmindustries.co.uk/coil-winding-tape-4b
We buy a similar stuff from a local rewinder, plus some thin card-like nomex based flexible insulation sheet for high voltage stuff.
That's also used to prevent wire insulation coming in direct contact with metal components, where it could chafe and short due to vibration.
Varnish impregnated cotton is fine, though, for what you need; it's just which you prefer.
Edit - you should really attach flying leads to the coil before final taping and varnishing. Carefully scrape or sand off the insulation from about a centimetre or so of wire, tin the bare ends and solder on a length of stranded, insulated wire.
Use heatshrink sleeving to insulate the soldered joints.
They should be positioned so they are within the final varnish & tape wrap of the coil, so only the flexible leads are exposed.
Doing that avoids any possibility of the coil ends breaking while you are working on it or later due to vibration; it only takes a few bends in the same are to crack the single-strand wire in the coil.
That type of wire is normally classed as having "solder though" insulation - but carefully scraping it to expose copper first is a lot less messy and smelly.. Any remaining tiny traces of insulation should melt away as it is tinned.