Charms don’t override/preempt each other unless that is called out in the rules
The quick answer is no — abilities or spells that impose the charmed condition don’t cancel previous charmed conditions. They can be in effect at the same time.
The answers to this question covered some of these charming rules
One exception is in the case of harpies, once a target is charmed by a given harpy, other harpies cannot charm the creature.
Charms are not all equal
Things get complicated because there is not a single “charm effect.” Various spells or abilities which impose the charmed condition (many of which are called “charm” or include the word in their name) actually impose very different sets of effects (See below).
For example, succubus can command a humanoid affected by her charm ability, while charm person only makes the target regard the caster as a “friendly acquaintance.” In general, the succubus’ commands would take priority over the request made with the benefit of Charm Person.
Charms in direct opposition
The exception to the above is if the command directly contradicts an effect of the other power, or contradicts the charmed condition, for example, if the succubus orders its thrall to attack a wizard who charmed the same humanoid. The charmed condition precludes this (see Appendix A: Conditions, PH, p.290).
In that case, the DM may rule that a contest is in order (See Contests, DMG, p 238) or may rule that the charmed condition precludes the action.
Crown of Madness vs. Vampire Charm
In the question’s particular case, the caster of Crown of Madness could maintain control the charmed character as long as the caster meets the requirements of the spell. That is, choosing a creature for the charmed PC to attack before moving on the first round (which might necessitate the PC attacking an ally) and “using his action to control the target” on subsequent rounds (PH p.230).
Also note the vampire’s effect might outlast Crown of Madness. The spell would persist while the caster’s concentration lasts, up to 1 minute. After that, the vampire’s charm, which lasts 24 hours, would still be in effect.
If the caster ordered the charmed PC to attack the vampire, then that would be a Contest as described above.
Differing Charm Effects
Here are a few charm effects for reference. There are a lot of subtle differences, so you have to be aware of the specifics to determine how they would interact.
Crown of Madness (PH p.229):
The charmed target must use its action before moving on each of its turns to make a melee attack against a creature other than itself that you mentally choose
Vampire (MM p. 297):
the target isn’t under the vampire's control
Charm Person (PH p. 221)
The charmed creature regards you as a friendly acquaintence.
Dominate Person:
While the target is charmed, you have a telepathic link with it...You can use this telepathic link to issue commands...
Succubus (MM p. 285):
The charmed target obeys the fiend’s verbal or telepathic commands.
Harpy (MM p.181):
While charmed by the harpy, a target is incapacitated and ignores the songs of other harpies.
The creature does not automatically became loyal
The spell description says the affected creature "can't attack the charmer". This doesn't mean it can't attack the charmer's allies. This doesn't mean it became loyal to the charmer — it might still consider the charmer hostile. The spell only says it can't attack the charmer, and the charmer has advantage on social interaction checks, nothing more. Of course, a DM can decide otherwise.
Speaking of social interactions, the DM has the final word — it is the DM who sets the DC, so a player might easily succeed automatically or fail despite of the advantage.
More about the general idea on how spells work in 5e - What is the source of the "spells do only what they say they do" rules interpretation principle?
Best Answer
The vampire’s bite is harmful, so it triggers a save against the charm.
The charmed creature gets to make a save “each time the vampire or the vampire’s companions do anything harmful to the target”. The vampire’s bite says:
This is obviously harmful. Being a willing victim does not make it not harmful. So the bite triggers a saving throw against the charm.
It should be noted that (by rules as written) being a willing victim makes a creature an eligible target for the bite attack, but it does not mean the bite hits automatically. The vampire still has to make an attack roll against the target’s armor class, though a DM could reasonably give the vampire advantage on the attack. In the event that the bite misses, no harm has been done to the target, so no saving throw against the charm would be triggered. However, it would not be unreasonable for the DM to rule that the bite is an automatic hit in some, or even most, cases. For example, I would still require an attack roll to hit a willing victim of a race with the natural armor trait whose neck would still present a challenge for the vampire’s teeth.