By which I mean, was there any kind of public health issue? Was there any percentage of the population going through withdrawals? And if so how was that handled? I did a small bit of poking around and I could easily find sources about there having been cocaine in the soda and why they stopped doing it but so far nothing I've found talks about the aftermath of removal.
Cocaine was removed from coca-cola 11 years before cocaine itself became illegal.
Despite Coca-Cola’s economic triumph, critics, many of them Progressive social reformers, still equated its refreshment with “dope” and associated Coke jingles with drug abuse. Critics also exploited Jim Crow fears, with sensational tales of “Negroes” hopped up on soda fountain binges...to preempt further negative publicity its famed president, Asa G. Candler, quietly withdrew cocaine from the product in 1903 (Gootenberg 2009, p.199 )
This means that while is was possible for any consumer of cocaine in coca-cola to get it from elsewhere, Spillane writes in his chapter in "Cocaine: Global Histories" that
Americans were offered new “de-cocainized” forms of Vin Mariani and Coca-Cola. In response, most of these consumers appear to have simply abandoned their use of cocaine – certainly there is no evidence that these users now adopted the use of the pure cocaine still available (Spillane 1999, p.40)
Nevertheless, the topic of pre-prohibition cocaine is an interesting one - as is the assumption that there would be public health issues related to cocaine (in coca cola or in general). The ingrained notion that certain drugs have objective "intrinsic" properties that are devoid of any social or cultural influences is one that is prevalent in contemporary western culture, and particularly frustrating to encounter in the field of neuroscience. Because of this, I'm going to take the opportunity in this answer to expound on the public health and cocaine use in the19th/early 20th century.
Sources on this topic are difficult to come by, partially because during this period the modern notions of drug abuse and use were still in their infancy, and partially because medical and social research was conducted in a different manner(Gootenberg, 1999 p. 3). For example, publications about cocaine abuse (cocainism) were largely case studies(Spillane 1999, p.40), and I am unaware of any large scale epidemiological studies conducted.
That being said, with respect to the health consequences of the large increase in legal cocaine supply (and implied consumption) at the turn of the 20th century, Spillane writes
Whether of not this increase in popularity was matched by a corresponding increase in health problems or cocaine abuse is more difficult to determine...On the one hand many (if not most) users appear to have consumed in moderation, experiencing few serious problems. On the other hand, there were also those for whom chronic heavy cocaine use exacted serious costs.(Spillane 1999, p.40)
It is likely that consumers of coke fell into the former category for two reasons. Firstly, route of administration is extremely important in determining the reinforcing effect of a drug; generally drugs that more rapidly reach effective concentrations in the brain are more reinforcing (Volkow et al. 2000, Samaha & Robinson 2005). The cocaine in coca-cola was consumed orally, one of the slower routes of administration. Secondly, the dosage of cocaine in coca-cola, although variable over time, was low (60-30mg per 8oz serving).
For some perspective, cocaine has a long history of use in and around the Andes, specifically in the form of the coca leaf. The pattern of use by Andean peoples is characterized by low doses (200-500mg per day) and a slow route of administration. It appears that the initial pattern of use when cocaine was adopted in North America and Europe was similar to this low dose pattern, and likely resulted in very few cases of "abuse". It was likely that only with the rise in popularity of faster routes of administration and access to purer cocaine that abuse started to occur, with daily consumptions in some cases reaching 12 grams - a dosage that would not be feasible with either coca leaves, or most cocaine containing beverages.(Siegel 1985) Despite this fact, Spillane notes that
Again, however, reports of cocaine’s abuse often made little distinction between various kinds of product
As a result the anti-cocaine prohibition movement made "no distinctions between high- and low-potency goods", and as a consequence "made safe, popular coca goods disappear from the market"(Gootenberg 2009, 198)
Sources