Over the past few years there have been many times where I have sat down to write a paper related to education for refugees, I go to write the opening paragraph to set the stage about how many refugee children are in school/out of school, and ask the same question: "where is it I go to look for data again?"
There never seems to be a simple "one place" answer. It seems that overall, regional and national data collection systems vary; sometimes international data is easier to find, but often it too has holes because it relies on regional or local systems that may be hampered by conflict or other structural challenges hindering data collection.
This list is not comprehensive, and is based on my limited research so far. If you have resources or thoughts to add, please feel free to comment or send them my way!
Global Level
GPE
From a global level, the Global Partnership for Education (GPE) produces some helpful data related to monitoring and evaluation related to inclusion and equity broadly in education. It also collects useful cross-country data about education in crisis situations, including data about education in countries affected by fragility and conflict, education for refugees across countries, and also has some more regional and country level reports as well.
Global Education Monitoring (GEM) Report
Though not always as useful as in 2019 for the field of refugee education, the 2019 GEM report focuses on migration and displacement, and therefore highlighted key issues and data points within and across countries globally. The report theme varies by year.
UNHCR/UNICEF Reports
Reports like Education Uprooted: For every migrant, refugee and displaced child, education (UNICEF), Turn the Tide: Refugee Education in Crisis (UNHCR), or Left Behind: Refugee Education in Crisis (UNHCR) often provide useful numbers that I can't seem to find elsewhere. For example, from the first report, that "in 2015, there were 27 million children out of school in 24 conflict-affected countries" (p.7). When I go to look at the endnote, it says "UNICEF data and analysis, 2017." In practice, I haven't found a way to obtain that data. It could be that by having a contact at UNICEF you might be able to email them and obtain data, I'm not sure. That is how Sarah Dryden-Peterson and I recently obtained country level data for the number/percent of refugee children enrolled in schools in multiple different countries. We were looking for these numbers across a few different contexts, and when we couldn't find data available online, emailed someone at UNHCR who was able to contact the country level offices for their data. Getting that data can take not only weeks, but months - and in some cases they may not have collected the numbers you are looking for. And, sometimes they might have the data, but only available in a language you can't read. At this point you then have to leverage your networks to find someone who you trust to translate for you.
UNHCR Population Statistics Database
This resource provides both country specific and across country data. While it doesn't tell you education enrollment numbers, it does, unlike many other resources, give you the ability to separate data out by year, country and AGE (not common), in addition to by sex if that is needed. Not only that, but the age distribution is not just lumped into an "under 18" category for children and youth, but you can search by 0-4, 5-11, 5-17, and 12-17 (however, I'm not sure how the overlap between those latter three categories is accounted for).
Regional Level
OECD
Though not necessarily a "region," the OECD does collect some data about refugee education in OECD countries, for example as published in their 2019 working paper on refugee education integration models and practices.
UNICEF
Sometimes depending on "situations," UNICEF may also have data and information repositories related to refugee and migrant children, for example here from Greece and Italy.
Eurostat
If you are looking at European countries and happen to be looking for categories Eurostat has data on, you are lucky! They tend to have data across multiple countries when they do collect data on something, for example the number of asylum and first time asylum applicants, or population by sex, age, migration status and citizenship. Similar databases may exist for other regions that I have not come across.
Country Level
In my experience, this has completely varied by country. My hunch (and based on my limited experience so far) is that knowing people who know people can be useful here.