It's going to give a slight edge over normal rolling. Say you roll 2d8. That's a nice triangular distribution, easy to calculate. Now say you roll 1d8 and call it x. Then your roll 2d8, but the minimum result is x + 1. That minimum result is going to push your result up a bit. As it turns out, it pushes the average roll up by 0.41015625.
Calculating it exactly for all of the possibilities would be a pain. So I programmed some simulations. Compared to a rounded up average, the WWN method is down about 3.7 hit points over 20 levels (assuming rolling for first level). On the other hand, the standard D&D method is 10 hit points behind the rounded up average over 20 levels. One difference is that standard D&D rolling is going to be 10 hit points behind regardless of the size of the hit die (the previous results in this paragraph are based on a d8 hit die). For the WWN method how much is lost compared to rounded up average depends on the size of the hit die. For example, a d10 hit die only loses about 2.3 hit points over 20 levels compared to rounded up average.