how to drive 1930s bakelite headphones
Modern headphones are often 30 ohms per channel, some high-end ones can be 600 ohms per channel. So this 62 ohms falls within that range so I do not see why you could not drive them with almost any headphone output.

The maximum signal will very likely be determined by the distortion the headphones introduce. Having said that, most portable devices with a battery supply cannot deliver much power in a 124 ohm (62 ohm + 62 ohm in series) load since that requires a higher voltage.

If that is problematic you could improve this by connecting the two drivers in parallel instead of in series like they are now.

I would not put too much effort in an amplifier for these headphones until you've tested if they're worth it with decent equipment. So I would simply test them with a standard headphone socket output from a decent "Hi-Fi" amplifier. Amplifiers that can drive proper speakers are capable of generating high enough voltages to drive this headphone beyond what it can handle without too much distortion.

If you really need to build something then I suggest one of the headphone amplifier modules you can get on eBay, here's an example.