Hi Artur,
Yes a 24 x 2 1/4 will fit, but looks very undersize for a vintage bike and will give a rather harsh ride. I had a couple of old 25 x 3 tyres (made in Australia!) and they were labelled "25 x 3 to fit 24 x 2 1/4 rim".
If you look back, this issue has been discussed before on this forum. The "problem" is that the EW rims are very similar in diameter to a 26x3 rim, but are narrower. A 26x3 tyre will probably fit on EW rim, but is it safe to use? I don't know the answer. Some manufacturers used safety bolts to fix the tyre to the rim (even on BE tyres) - perhaps this could be a safe option? I suspect Andy is right, and that 26 x 3 rims might be the best and safest way to go.
Leon