Question: If one's gums bleed but blood does not come out of one's mouth and remains inside, does it nullify one's wudu'?
ANSWER
One's wudu' remains valid unless blood comes out of one's mouth. It is written in all of the books of fiqh that wudu' is nullified when blood comes out of the mouth or when one vomits blood. No fiqh book says that swallowing blood breaks wudu'.

It is written in the book Endless Bliss:
Inside the mouth, in terms of an ablution [wudu'], is deemed an inner limb, but it is considered an external limb when one is in a state of fasting. That is the reason why the blood issuing from a tooth or a wound inside the mouth while remaining in the mouth does not break an ablution. But after coming out of the mouth, if the blood is more than the spittle it breaks an ablution. Thick blood coming down from the head does not break an ablution even if it is more than the spittle. If the blood issuing from the stomach or from the lungs is thin, it breaks an ablution even if it is less than the spittle according to Shaikhayn (Imam-i A’zam Abu Hanifa and Imam-i Abu Yusuf). If one sees blood on the thing one bites into, one’s ablution will not be broken. If one sees blood on the miswak or on the tooth pick, this will not break one’s ablution if the inside of the mouth has not been smeared with blood. But, on the other hand, it will break one’s ablution if one puts one’s finger on the suspected part in one’s mouth and then sees blood on one’s finger [because one, by doing so, has taken the blood out of one's mouth]. (The section dealing with the nullifiers of wudu')

If blood accumulating in the mouth is not a mouthful, neither its accumulation nor swallowing it nullifies your ablution or namaz [salat]. So is the case with vomiting. (The section dealing with the disliked acts of salat)

The author of the book Bahr-ur raiq states, “In some cases the mouth is thought of as an internal part of the body. Hence, if a fasting person swallows his saliva, his fast will not break. It is like something dirty inside the body passing from stomach to intestines. Bleeding from an injury in the mouth, from taking a tooth out, or at the point where an injection was made, or blood coming from the stomach to the mouth doesn’t break a fast or an ablution. When one spits out or swallows this blood, if the saliva is greater than the blood, that is, if it is yellow in color, they are still not broken. It is the same when other things come to the mouth from the stomach, in which case neither the ablution nor the fast is broken. If a mouthful (comes to the mouth and) goes out of the mouth, both are broken. The inside of the mouth is sometimes considered to be an outer part of the body. The fast is not broken when water is taken into the mouth.” The same is noted in Jawhara, too. Hence, it is seen that, when a tooth is extracted, if there is much bleeding, the fast is not broken when one spits it out. When one is not fasting, one’s ablution is not broken when one swallows it. (The section dealing with the fast of Ramadan)

It is written in Durr-ul-mukhtar:
Vomiting a mouthful of bile, black blood clots, food, or water breaks wudu'. [Their coming out of the mouth breaks wudu'.] (The section dealing with wudu')

What breaks wudu' is filthy substance's coming out of the body, not filthy substance itself. Passing gas breaks wudu' not because gas is considered impure (najs) but because it exits from the place of impurity (najasat). (Ibn Abidin)

It is understood from these statements that blood and other impure things do not break wudu' unless they come out of the body. The wudu' of a person with hemorrhoids bleeding will not be nullified unless blood oozes out. To decide whether one's wudu' has been nullified or not, the criterion to be used is whether impurities (najasat) have come out of the body.

It is written in Durar wa'l Ghurar:

Wudu' is nullified if blood or pus predominates over saliva or equals it. It is not nullified if saliva predominates. (The section dealing with the nullifiers of wudu')

[What is meant by these statements is blood's coming out of the mouth. No fiqh book says that swallowing it breaks wudu'.]

Swallowing the blood in the mouth breaks neither wudu' nor salat. It is written in al-Fatawa al-Hindiyya:

If one, while performing salat, swallows the blood in between one's teeth, it does not break one's wudu', even if the blood predominates over the saliva. While performing salat, if one, deliberately or absent-mindedly, eats or drinks something, one's salat becomes nullified. However, if one who is offering salat swallows the blood oozing from in between one's teeth, one's salat is not nullified if the blood is not a mouthful. The same is written in the books Fatawa al-Kadihan and Khulasa. (The section dealing with the nullifiers of wudu')

If the blood coming out of the mouth equals saliva or predominates over it, wudu' is nullified. (Halabi as-saghir, Ni'mat-i Islam) [What is mentioned here is blood's coming out of the mouth, not its accumulation in the mouth.]

What breaks wudu' is not the accumulation of blood in the mouth but its coming out of the mouth. As the inside of the mouth, in terms of wudu', is deemed an inner limb, bleeding from gums will not break wudu' unless blood comes out of the mouth. It does not break wudu' either even when one swallows the blood.