the fluctuating tempo is due to a poor approach of OP and proves or disproves nothing regarding web audio. that stuff can be as steady as you want.
I also tried to minimize any visual events that cause a repaint on the main instrument screen.
samples_to_tick=Math.floor( (125.0/bpm) * (1.0/50.0)*sampleRate );
Upon the counter reaching zero, the render loop actually jumps to do sequencing and effects/modulation and then resets to counter.
In my case, the tick occurs at 50Hz because that's the PAL frame clock used in Amiga and MS-DOS module trackers like Scream Tracker and FastTracker 2. Typically trackers use a "speed" setting to additionally specify the number of ticks per pattern step - usually 6.
Although you do get sample-accurate timing, you also get latency up to the size of a single audio buffer (I use 4096 samples). While this isn't really an issue when doing purely playback, it obviously is when the audio is controlled real-time (eg. MIDI).
Anyway - the module player is at http://mod.haxor.fi/ and the source code at https://github.com/jhalme/webaudio-mod-player