This is really a two-part question, the second part of which probably isn't answerable.
If everyone were vegetarian, what would the impact on mortality be?
The BBC cites "Analysis and valuation of the health and climate change cobenefits of dietary change" by Springmann, Marco, H. Charles J. Godfray, Mike Rayner, and Peter Scarborough, in Proceedings of the National Academy of Sciences volume 113, no. 15 (2016).
The article provides the following detail:
Compared with the reference scenario, we project that adoption of global dietary guidelines (HGD) would result in 5.1 million avoided deaths per year [95% confidence interval (CI), 4.8–5.5 million] and 79 million years of life saved (CI, 75–83 million) (Fig. 1A and SI Appendix, Fig. S2). The equivalent figures for the vegetarian (VGT) diet are 7.3 million avoided deaths (CI, 7.0–7.6 million) and 114 million life years saved (CI, 111–118 million) and for the vegan (VGN) diet 8.1 million avoided deaths (CI, 7.8–8.5 million) and 129 million life years saved (CI, 125–133 million).
So they aren't saying that there would be hundreds of millions more people in the world, but that there would be five to eight million less people dying per year.
It's important to note that the results are broader than simply "not eating meat", which they point out in the discussion:
We found that about half of the global avoided deaths occurred because of the consumption of less red meat and that the other half was due to a combination of increased fruit and vegetable consumption and reductions in total energy intake (and the associated decreases in the fraction of people overweight and obese)
So simply eating healthier in general would reduce mortality rates -- you can eat less red meat, more fruits and veggies, and less total calories, without being vegetarian.
How would this impact the global population?
That's the question that's much harder to answer. It isn't addressed in the article, but I found the following statement in an article titled Mortality-Fertility Relationships (emphasis added):
It is axiomatic that, once death rates in a population have fallen steeply and irreversibly, birth rates must eventually follow. The alternative is rapid population growth that is unsustainable in the long term. However, the characterization of the link between mortality and fertility in classic statements of demographic transition theory has differed. [...]
Since these early theoretical contributions, scientists have accumulated a great deal of empirical evidence on the mortality—fertility link at the societal and family level. It has become clear that the relationship is not a simple mechanical one. Though it remains true that mortality decline has always preceded secular declines in fertility, the degree of prior mortality improvement, the absolute level and age pattern of mortality at the onset of fertility decline, and the time lags involved vary widely among societies.
Basically, while the immediate impact of less people dying is more people in the world, the long-term impacts of reduced mortality (less people dying) on fertility (how many people are born) varies greatly due to a lot of factors that aren't fully understood.