Do histones have to be synthesized before DNA is replicated to allow the DNA to coil around histones?
Yes, they have to. But that is just half of the story.
The (canonical) histones which are used in DNA replication are synthesized at the beginning of the S phase, and subsequently transported into the nucleus. Studies have shown that newly synthesized DNA is immediately packed into nucleosomes. Thus, it is necessary that these structures are available prior to (or at least just in time with) the replication.
However, there are different models on how the histones are inserted into new DNA. One model assumes that old and new nucleosomes are both incorporated into newly synthesized DNA after the replication fork. Others assume a semi-conservative approach where the old histones are disassembled into their subunits and then mixed together with the newly synthesized ones. For example, there is a model that proposes that the parental H2A/H2B and the H3/H4 dimers disassociate, while another assumes a disassociation of the H3/H4 dimers as well.
a Angélique Galvani and Christophe Thiriet (2013). Replicating – DNA in the Refractory Chromatin Environment, The Mechanisms of DNA Replication, Dr. David Stuart (Ed.), InTech, DOI: 10.5772/52656. Available from: https://www.intechopen.com/books/the-mechanisms-of-dna-replication/replicating-dna-in-the-refractory-chromatin-environment
In Molecular Biology of the Cell (Chapter 4), it is written that
The major histones are synthesized primarily during the S phase of the cell cycle and assembled into nucleosomes on the daughter DNA helices just behind the replication fork (see Figure 5–32). In contrast, most histone variants are synthesized throughout interphase. They are often inserted into already-formed chromatin, which requires a histone-exchange process catalyzed by the ATP-dependent chromatin remodeling complexes discussed previously