If you ever have any questions about how the Turkish language changed in the first four decades of the 20th century, you should drop everything and read Turkish Language Reform: A Catastrophic Success by Geoffrey Lewis.

The short answer is no, not everyone had to relearn how to read and write (emphasis on that word).

First, the reality was that to begin with the literacy rate was not particularly high in the Ottoman Empire (well under 50% of the Turkish population), so for most citizens of the Turkish Republic this was the first time they were going to be learning to read (if they bothered to learn at all). To help these new learners, the law was passed at the same time as the republic's new education system was spreading across the country - the picture of Atatürk teaching the alphabet is famously used as an example of this.

Second, many well-educated Ottomans would have been familiar with the Latin alphabet anyhow due to the prevelance of French as an international language. I wish we had statistics on the percentage of literate Turks who knew French or another European language, but it wasn't insignificant judging by what intellectual life was like in the late Empire. Of course these intellectuals would have had to learn the new letters or letter-to-sounds mappings that were created for modern Turkish but the orthography is honestly quite simple and regular.

So when you take the population as a whole and remove those who already knew the Latin alphabet and then subtract those who couldn't read at all in any alphabet, you are left with a fairly small percentage of people who only knew the Arabic alphabet and therefore had to relearn how to read and write.

I should also note that most opposition to the adoption of the Latin alphabet was on ideological grounds rather than practical ones. I feel comfortable stating that the Latin alphabet is objectively better suited for writing Turkish than the Arabic alphabet. The classic examples of why include that standard Istanbul Turkish has 8 "native" vowel sounds plus three borrowed long vowels, but Arabic has only three vowels to write them with. It's no better when you get to the consonants, Arabic has three letters that Turks pronounce h and three more that they pronounce s (similar for d, t, z, etc) and someone literate in Ottoman Turkish would have to memorise when to use which. And on top of this some Arabic letters, like ك had to run triple duty when writing Turkish words and could sound like k, n or y depending on context.

There is very little orthographic ambiguity left in modern Turkish's Latin alphabet - although it's not perfect (I'm looking at you ğ, k and l!) - and that certainly helped to improve literacy rates in Turkey.

As a footnote, Lewis makes the case that the aggressive changes to the language in the 1930s were far more dramatic and problematic than the alphabet switch in the 20s. Thousands of words and grammatical patterns were stripped out of the Ottoman language with thousands of new terms being created out of thin air; this is when you see the real communication issues as people struggled with the constant stream of neologisms that the "reformers" in Ankara were churning out.