Generally, sample sizes for registries are predicated not upon a pre-specified null hypothesis, as you would have in a clinical trial, but on the tension or equilibrium between budgets, enrollment timelines, total running time for the registry, and the desire to achieve a reasonable level of precision (not power) in the estimation of key parameters. The latter, perhaps, being the easiest part of the process.
The general term of art, is that you will have a “convenience sample” of some pre-defined, target size.
In a multi-site, much less a multi-country setting, there is the question of whether or not the data within individual countries are representative of that country’s experience, and then, how is the overall aggregate dataset biased by the contributions of the subsets. Do you cap the participation of any site or country, so as to not materially bias the aggregate towards the subset, and if so, at what level, or do you allow each to enroll as many patients as they can, within your enrollment timeline, to increase the likelihood of hitting your enrollment target?
The key issue in the above, is whether or not you will be able to generalize your findings to populations outside of what you have in the registry, or are those findings idiosyncratic of the sample in your registry.
Also, bear in mind that enrollment is generally the biggest challenge for any study.
Part of the pre-study planning process is to get some reasonable estimates of the prevalence of the underlying disease in the populations of interest, get a sense for per site volumes in each geographic area, and then make some presumptions of how many patients can be enrolled at each site, per unit of time, within your overall enrollment time frame.
If you already have a finite budget that you have to work within, you need to model how all of those assumptions fit within the budget, and adjust accordingly. If you are first looking at creating a budget, then you need to model those assumption to conform to your research goals, and estimate a budget based upon those factors. You also need to consider the varied regulatory issues in each country, and what that means operationally and financially for your registry.
It will typically take a multi-disciplinary, team approach, to move towards defining all of these parameters so that you end up with a reasonable design, target sample size, and a protocol that reflects these various factors.