Looking at the film I wondered if a vampire bitten by Blade would become a daywalker, or whether biting a vampire would cause Blade to turn into a full vampire, or would nothing at all happen?
Blade is a dhampir. That means he is a half-vampire. Blade CAN be turned into a full vampire. This has come up in the comic books many times and was even discussed in the very movie that TV series spun from.
Usually in Marvel comics a vampire has to drain you and then feed you some of his own blood to work the change.
As for Morbius, yes, Morbius was created with science instead of sorcery or a vampire bite but his vampirism works very much like the supernatural version (just no need for an invitation to enter a home, religious symbols have no affect on him, and he cannot turn into an animal the way Dracula can). However Morbius can make other people into vampires the traditional way.
As far as I know, the Blade series has never really delved into the concept vampiric cannibalism. However, from what I understand of the Blade universe's version of vampirism, it's disease based rather than supernatural.
If we consider the TV series of Blade (its one season, that is) as canon, that would mean Blade's unique traits are differences in how the vampirism disease is expressed, rather than an actual mutation in the strain. There are two pieces of evidence to support this from that series: -The "Bad Bloods" gang that took Eric/Blade in as a child after running away from his father. Blade fed on them, thus turning them into vampires. When they later kidnap him in daylight, they are seen wearing heavy protective gear much like that used by motorcyclists. Given their tactics to neutralize Blade (driving into him with a large vehicle), they're not overly concerned about being covert, which suggests they're shielding themselves from the sunlight. Later in that same story sequence, Blade's normal weapons are just as effective against the Bad Bloods as any other vampire. Therefore, they are not daywalkers despite being sired by a daywalker. -Similarly, the episode deliving into Marcus's backstory shows him being turned over to a clan of Native vampires who are just fine in the daylight. While Marcus is never actually shown to be vulnerable to the silver/garlic/UV weapons that Blade uses, he does demonstrate the same aversion to these threats that other vampires do.
This indicates that being a daywalker does not modify the disease itself; it is how the change is expressed in a fetus rather than a child (Charlotte, also from the series) or adult. Thus, Blade and his foes carry the same strain of the vampirism disease. In the case of normal diseases, a person who is exposed to the same illness they're already suffering from will not get any sicker. (The same would not necessarily be true of 2 different diseases, though some more serious diseases (HIV, for instance) often have multiple strains infecting the same patient, as the virus evolves rapidly to survive the various treatments.
Furthermore, the series does contain a couple of references to Blade Trinity (Blade's escape from the psychiatric ward, for instance), so it is safe to say that the series is in the same continuity as the film trilogy. I don't know of any examples in the film trilogy itself to support this analysis.
Therefore, I would conclude that a vampire feeding on Blade or vice versa would have no effect on either one's form of vampirism.
However, Blade is a Marvel property and thus the comic form is the true canon of the story. As I am not familiar with the comic version of Blade, my conclusion may not apply to this version of the Blade story.
According to the Marvel wiki blade is a dhampir and it more or less implies the vampirism is supernatural disease. Here is the link for your reading pleasure.
Also it seems can sometimes be changed by bites from certain vampires, such as after being bitten by Morbius the Living Vampire his powers increase. Although, Morbius isn't a normal vampire; he's a human that accidentally made a vampiric disease and is trying to fix himself. For more information on the vampires read the issues regarding Dracula as well.
So, to answer your question, in the comics, if Blade bites someone, it's up to the writer if they turn into a daywalker or not, but as far as I know, this hasn't happened yet in the comics.