At minimum, you'll need a working Bluetooth speaker with good battery life and a suitable portable radio. This would be bare minimum though and I recommend a few more items.
For my build out, I decided I wanted long battery life and improved audio. For the battery, I used a 3.7v lithium extended capacity phone battery. This battery spec'd out at 3500mAh versus the 500mAh battery that was part of my donor Bluetooth speaker. This should give me about 10 hrs playtime between charges at reasonable volume. The stock Bluetooth speaker battery was good for maybe 2 hrs or so.
For the speaker, I used a driver yanked from an old surround system. This speaker had a high compliance cloth roll surround. This will give me more low end and fuller sound. Foam surround speakers are much more common, but all eventually fail due to foam rot. I'm not a fan of foam surround speakers for this reason. I recommend going with anything other than a foam surround driver.
Why not reuse the original radio speaker?
You can, but chances are it won't last very long and it will sound squawky. It was designed for half a watt or less of audio and not really meant for full range music. Save yourself twice the work and just get a replacement speaker. Even the cheapest Bluetooth speaker amps easily put out more power than the old radio speaker can handle.
A word about donor radios..
While most portable tube radios don't get much love in the collector world, early transistor portables do. There were some lunchbox sized early transistor portables that at first glance may look like an old tube radio. These are obscure and hard to come by as they were really expensive in their day. If you do find one, don't hack it. Sell it off on eBay and you should have enough cash to buy two or three tube portables.
If you find a Zenith transoceanic tube portable, those are valuable as well. The very early ones from before WWII can bring hundreds of dollars on eBay. Don't hack a Zenith transoceanic!