Effects of using Horlicks original malt in bagel recipe

Yes, definitely! It may darken your product a little but the flavor should be right. Non-diastatic malt powder or malt syrup would be preferred, but you can use Horlick's in both your dough and the boiling water. I also use Malta (India or Goya) drinks in a pinch. Note that Horlick's has some milk powder in it (about 14%). That's where the possible darkening of your bagels could come from.

You could try throwing a little in the boiling water, but I wouldn't recommend adding it to the bagel itself because it might cause some funky flavors.
Here's a recipe that only uses malt in the boiling: https://www.seriouseats.com/recipes/2017/01/homemade-bagels-recipe.html