Measure from square taper end to end. Ignore threaded. Hal fords have a
118mm BB for a tenner which would be fine for this bike which will get little use. ETA: Spa Cycles have a wide range: eg
Stronglight BB.
Reassembly of current cup&cone:
Clean everything. Toothbrush for the BB shell threads and paraffin (or whatever) in a cup to get the caged bearings clean.
Replace the now heavily regreased caged bearings into both the right and left side cup; and screw the right one in tight (reverse thread).
Grease on both cones of the spindle and insert so cup is up against bearings.
Screw the left hand cup in till resistance, and then screw on lockring gently tight.
Check spindle cannot move in and out. Check spindle spins smoothly.
Adjust compression till above criteria optimised (loosen lockring, turn cup as nec, tighten lockring, check).