In the (beautifully rendered) video you linked to, the green molecules are DNA polymerases. So you can already see that there are more than two DNA polymerases at work!
At each replication fork, there is generally one DNA polymerase working on the leading strand, but on the lagging strand, multiple DNA polymerases may be working at the same time (as depicted in the video). Note that different types of polymerase are thought to be primarily responsible for the leading (Pol ε) and lagging (Pol δ) strands in eukaryotes (but not prokaryotes, which use Pol III for both).
Next, note that at every origin of replication, there are two replication forks working simultaneously, one in each direction. So that's double the number of polymerase molecules.
On top of that, a single eukaryotic cell has multiple chromosomes. Each one must have at least one origin of replication of its own, which can operate in parallel. In reality, each eukaryotic chromosome has up to thousands of origins of replication (prokaryotic chromosomes generally only have one). Human cells have on the order of 100,000 origins of replication. So depending on how many origins are active simultaneously, there are likely many thousands of polymerase molecules acting at once to replicate the DNA of a single eukaryotic cell.
You could probably do a rough estimate of the number of polymerase molecules active simultaneously by taking the average duration of S phase in your cell of interest and dividing by the genome size and the speed of polymerase.