I went to a recorder workshop at Whitby and we were told that it was not worth buying a cheap (i.e. under £100) wooden recorder as a softwood instrument would not last, and they would not be consistent. Some yours ago I bought a descant recorder for my son to learn at school (plastic mouthpiece, wooden body) and found that it was unplayable. The recommended makes are Aulos and Yamaha, and I certainly found that when I changed to an Aulos the second octave was easier to play.Not the cheapest but under £10, can you go wrong.
I like the sound of my wooden Treble (a 21st present many years ago, but its not much use in sessions)