You need at least 2 seriesed 1N4007 for each leg of the HT winding. Current handling of the 1N4007 is more than ample, but the problem is the reverse voltage you'll encounter in this application, so my suggestion of using 2 1N4007 in series.
Anyway, solid state diodes in vintage radios
Might be ok as a test, but I would never ever consider this as a definitive solution, unless the rectifier is unobtanium, and even then I would think of an alternative tube to swap.
This is a very interesting fault with this 5V winding, too bad you are not in Italy, otherwise I would fix this for you in no time flat. The melted wax indicates the transformer ran warm in it's life, probably not helped by bad paper and 'lytic caps leaking like sieves!
Cheers
Giulio