Tellytech your reactor is a magnox and is now out of date.
As for the lighting ballast I wouldnt take the original windings into account, a better approach would be to use the universal transformer equation and calculate the primary turns based on the material (iron, or it could be silicon steel), frequency, cross sectional area and voltage.
One thing to bear in mind is the contruction of the ballast, I see all the I's and E's are aligned, this is a high magnetic leakage core, meaning saturation and switch on surge could be an issue, maybe not a problem if it is switched on under load and not on for long periods.
This is an interesting vid for transformer design, ok its for torroids, but it gives a non mathematical approach to transformer design.
Wikipedia has a good article, the universal transformer equation is about 1/4 the way down, laminated transformers usually saturate at 1.o Tesla, so that would be your Bpeak.
Adding 30% extra winding turns is a good idea.
https://en.wikipedia.org/wiki/Transformer
Just a semi calculated rough guess I'd say you need around 800 turns primary (based on a core 3x5cm centre), so the secondary would be around 40 turns.
You could join both transformers together, and wind the windings around them both as if they were one, however there would be a lot of leakage unless you bolted or welded them together very tight, so I wouldnt reccomend this, just use the one.