Measure delay of sound in wood with piezo sensor without an oscilloscope
Rather than building counters and threshold devices, if your software foo is up to it and you have a PC, then using a sound card oscilloscope or a recording application like Audacity (which can show you waveforms just like a 'scope) could give you all you need. Building a thresholding device to get a nice logic edge from the sensors is a job that will probably need an oscilloscope to be successful.

At most, all you need is a pre-amplifier per channel, but with a sufficiently sensitive input, even this might not be required. Connect the two sensors to the two input channels.

As soon as you hook the sensors up to the PC, you can make a recording and see what you have.

As you would be recording analogue waveforms, your temporal resolution is not limited to the sampling rate. Programs like Audacity allow you to shift time waveforms with sub-sample resolution. Shift one waveform until it lines up with the other, and the shift you needed is the answer. Often PC hardware is capable of higher rates than 48kHz, 192k is not uncommon.

Using an existing PC and a free program like Audacity is about as low budget as you can get.