I ended up using canvas.toDataURL instead of trying to convert this myself.

If your data may contain multi-byte sequences (not a plain ASCII sequence) and your browser has TextDecoder, then you should use that to decode your data (specify the required encoding for the TextDecoder):

var u8 = new Uint8Array([65, 66, 67, 68]);
var decoder = new TextDecoder('utf8');
var b64encoded = btoa(decoder.decode(u8));

If you need to support browsers that do not have TextDecoder (currently just IE and Edge), then the best option is to use a TextDecoder polyfill.

If your data contains plain ASCII (not multibyte Unicode/UTF-8) then there is a simple alternative using String.fromCharCode that should be fairly universally supported:

var ascii = new Uint8Array([65, 66, 67, 68]);
var b64encoded = btoa(String.fromCharCode.apply(null, ascii));

And to decode the base64 string back to a Uint8Array:

var u8_2 = new Uint8Array(atob(b64encoded).split("").map(function(c) {
    return c.charCodeAt(0); }));

If you have very large array buffers then the apply may fail with Maximum call stack size exceeded and you may need to chunk the buffer (based on the one posted by @RohitSengar). Again, note that this is only correct if your buffer only contains non-multibyte ASCII characters:

function Uint8ToString(u8a){
  var CHUNK_SZ = 0x8000;
  var c = [];
  for (var i=0; i < u8a.length; i+=CHUNK_SZ) {
    c.push(String.fromCharCode.apply(null, u8a.subarray(i, i+CHUNK_SZ)));
  }
  return c.join("");
}
// Usage
var u8 = new Uint8Array([65, 66, 67, 68]);
var b64encoded = btoa(Uint8ToString(u8));

I'm guessing your base64-encoded URI is actually a PNG, so you could do the following:

const {PNG} = require('pngjs');
const jsqr = require('jsqr');

const dataUri = 'data:image/png;base64,ABCDEFYOURIMAGEHEREABCDEF';
const png = PNG.sync.read(Buffer.from(dataUri.slice('data:image/png;base64,'.length), 'base64'));
const code = jsqr(Uint8ClampedArray.from(png.data), png.width, png.height);

// code.data now contains the URL string encoded by the QR code

Since you don't want to use Canvas, you'll need to implement encoding/decoding for various image types yourself. Your code will look something like:

function dataUri2Array(uri){
    b64 = uri.slice(uri.indexOf(',')+1);
    type = getImageType(uri); // *
    data = atob(b64);
    switch (type) {
    case "jpeg":
         arr = decodeJpeg(data);
         break;
    case "gif":
         arr = decodeGif(data);
         break;
    case "png":
         arr = decodePng(data);
         break;
    default:
         break;
    }
    u = new Uint8ClampedArray(arr);
    return u;
}

The getImageType(uri) function and all the decodeX(data) functions need to be implemented. Here are some libraries that might help: jpeg-js, node-pngjs, bmp-js, etc. Getting the pixel arrays from arbitary data is nontrivial, especially when dealing with non-raw types (i.e. BMP).

If you want to write the decompressors yourself, the Wiki page may be a good introduction, but I also suggest reading up on Shannon-Fano and other kinds of encoding methods to familiarize yourself with the concept.

I hope you have a good reason for not using Canvas.