Are Seraphim and Cherubim angels?

According to The New Unger's Bible Dictionary, definition of Cherub

The cherubim seem to be actual beings of the angelic order.

and Seraphim

An order of celestial beings, whom Isaiah beheld in vision standing above Jehovah as He sat upon His trhone (Isa. 6:2, 6).

So, cherubim are likely beings of angelic order whereas seraphim are an order of celestial beings. While an angel is a celestial being, a celestial being might not be an angel.

Based solely in the definitions above, we can't be 100% they are angels, even though cherubim seem to be.

Are Seraphim and Cherubim angels?

The short answer is yes.

Angels appear on numerous occasions in both the Old Testament and the New Testament. However, the Scriptures do not always reveal their angelic rank on each occasion. Traditionally, there are Nine choirs of angels. This question deals with the biblical basis of the two highest choir of angels: the Seraphim and the Cherubim.

The Apostle St. Peter dealt with angels when he was freed from the prison of King Herod Agrippa. You can read about St. Peter’s Miraculous Escape From Prison here.

St. Paul was encouraged by an angel, just before his shipwreck on Malta (Acts 27: 21-24)

An angel sometimes would stir the waters in the pool of Bethesda.

4 For an angel went down at a certain season into the pool, and troubled the water: whosoever then first after the troubling of the water stepped in was made whole of whatsoever disease he had. - John 1: 9

Thus we can conclude that angels do exist.

Now let us see if there are any references to Seraphs and Cherubs being angels in the Scriptures.

St. Paul definitely believe in an angelic hierarchy.

16 For in him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things have been created through him and for him. - Colossians 1:16

For we wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places. - Ephesians 6:12

Cherubs are the most commonly name angel beings named in the bible.

The cherubim are the most frequently occurring heavenly creature in the Hebrew Bible, with the Hebrew word appearing 91 times. The first occurrence is in the Book of Genesis 3:24. Despite these many references, the role of the cherubim is never explicitly elucidated. While Hebrew tradition must have conceived of the cherubim as guardians of the Garden of Eden (in which they guard the way to the Tree of life), they are often depicted as performing other roles; for example in the Book of Ezekiel, they transport Yahweh's throne. The cherub who appears in the "Song of David", a poem which occurs twice in the Hebrew Bible, in 2 Samuel 22 and Psalm 18, participates in Yahweh's theophany and is imagined as a vehicle upon which the deity descends to earth from heaven in order to rescue the speaker (see 2 Samuel 22:11, Psalm 18:10).

In Exodus 25:18–22, God tells Moses to make multiple images of cherubim at specific points around the Ark of the Covenant. Many appearances of the words cherub and cherubim in the Bible refer to the gold cherubim images on the mercy seat of the Ark, as well as images on the curtains of the Tabernacle and in Solomon's Temple, including two measuring ten cubits high.

In Isaiah 37:16, Hezekiah prays, addressing God as "enthroned above the cherubim" (referring to the mercy seat). In regard's to Solomon's Temple as described in 1 Kings, Eichler renders the phrase yoshev ha-keruvim as “who dwells among the cherubim”. This phrase is the same in 1 Kings and Isaiah. Eichler's interpretation in contrast to common translations for many years that rendered it as “who sits upon the cherubim”. This has implications for the understanding of whether the ark of the covenant in the Temple was literally YHWH's throne or simply an indicator of YHWH's immanence.

Cherubim feature at some length in the Book of Ezekiel. While they first appear in chapter one, in which they are transporting the throne of God by the river Chebar, they are not called cherubim until chapter 10.[21] In Ezekiel 1:5–11 they are described as having the likeness of a man, and having four faces: that of a man, a lion (on the right side), and ox (on the left side), and an eagle. The four faces represent the four domains of God's rule: the man represents humanity; the lion, wild animals; the ox, domestic animals; and the eagle, birds. These faces peer out from the center of an array of four wings; these wings are joined to each other, two of these are stretched upward, and the other two cover their bodies. Under their wings are human hands; their legs are described as straight, and their feet like those of a calf, shining like polished brass. Between the creatures glowing coals that moved between them could be seen, their fire "went up and down", and lightning burst forth from it. The cherubs also moved like flashes of lightning.

In Ezekiel chapter 10, another full description of the cherubim appears with slight differences in details. Three of the four faces are the same – man, lion and eagle – but where chapter one has the face of an ox, Ezekiel 10:14 says "face of a cherub". Ezekiel equates the cherubim of chapter ten with the living creatures of chapter one: "They were the same creatures (חיה) I had seen by the river Chebar" (Ezekiel 10:15) and "These were the living creatures I had seen under the God of Israel on the banks of the river Chebar" (Ezekiel 10:20). In Ezekiel 41:18–20, they are portrayed as having two faces, although this is probably because they are depicted in profile. - Cherub (Wikipedia)

As for Seraphs, they are named less frequently in Scripture, but nonetheless are of the angelic Orders.

There is emerging consensus that the motifs used to display seraphs in Hyksos-era Palestine had their original sources in Egyptian uraeus iconography.

The word saraph/seraphim appears three times in the Torah (Numbers 21:6–8, Deuteronomy 8:15) and four times in the Book of Isaiah (6:2–6, 14:29, 30:6). In Isaiah 6:2–6 the term is used to describe a type of celestial being or angel. The other five uses of the word refer to serpents.

The vision in Isaiah Chapter 6 of seraphim in an idealized version of Solomon's Temple represents the sole instance in the Hebrew Bible of this word being used to describe celestial beings. "... I saw also the Lord sitting upon a throne, high and lifted up, and his train filled the temple. Above it stood the seraphims: each one had six wings; with twain he covered his face, and with twain he covered his feet, and with twain he did fly." (Isaiah 6:1–3) And one cried to another, "Holy, holy, holy, is YHWH of hosts: the whole earth is full of His glory." (verses 2–3) One seraph carries out an act of ritual purification for the prophet by touching his lips with a live coal from the altar (verses 6–7).

The text describes the "seraphim" as winged celestial beings with a fiery passion for doing God's good work.8 Notwithstanding the wording of the text itself, at least one Hebrew scholar claims that in the Hebrew Bible the seraphim do not have the status of angels, and that it is only in later sources (like De Coelesti Hierarchia or Summa Theologiae) that they are considered to be a division of the divine messengers.

Seraphim appear in the 2nd-century BC Book of Enoch, where they are mentioned, in conjunction with cherubim, as the heavenly creatures standing nearest to the throne of God. They are also called the Akyəst (Ge'ez: አክይስት "serpents", "dragons"; an alternate term for Hell).

In the Second Book of Enoch, two classes of celestial beings are mentioned alongside the seraphim and cherubim, known as the phoenixes and the chalkydri (Ancient Greek: χαλκύδραι khalkýdrai, compd. of χαλκός khalkós "brass, copper" + ὕδρα hýdra "hydra", "water-serpent" — lit. "brazen hydras", "copper serpents"). Both are described as "flying elements of the sun" that reside in either the 4th or 6th heaven, who have twelve wings and burst into song at sunrise.

In the Book of Revelation (4:4–8), the beasts are described as being forever in God's presence and praising him: "[A]nd they rest not day and night, saying, 'Holy, holy, holy, Lord God Almighty, which was, and is, and is to come." This account differs slightly from the account of Isaiah, stating in the eighth verse, "And the four beasts had each of them six wings about him; and they were full of eyes within". They appear also in the Gnostic text, On the Origin of the World. - Seraph (Wikipedia)

There can be little doubt that both Seraphims and Cherubims are angels.