Angband Orcish (often abbreviated as “AO”) is an unofficial name for language spoken by orcs of Morgoth prior to creation of Black Speech in the 2nd Age. The term was probably invented by Anthony Appleyard, a member of the Tolkien Society and contributor to Quettar journal. Angband was the citadel constructed by Morgoth during Years of the Trees, and Sauron was the commander there. So it's possible that some words of 1st age orcish were borrowed into his Black Speech, as the name Boldog appeared both in 1st and 3rd age. However, in Tolkien's sources this language was called just “orcish” or “Orquin”:

Orquin, or Orquian, the language of the Orcs,
the soldiers and creatures of Morgoth,
was partly itself of Valian origin,
for it was derived from the Vala Morgoth.
But the speech which he taught he perverted
wilfully to evil, as he did all things,
and the language of the Orcs was hideous
and foul and utterly unlike the languages of the Qendi. 

    (HOME vol. 5, The Lost Road and Other Writings,
    Part Two, Chapter V: The Lhammas)

“The Lhammas” chapter of “The Lost Road and Other Writings”, a fictional essay of Pengolod of Gondolin, also contains the following lines on languages of men and borrowing from Orcish:

The languages of Men were from their beginning diverse and various;
yet they were for the most part derived remotely
from the language of the Valar.
<...>
But other Men learned also wholly or in part
of the Orcs and of the Dwarves;

But it also stated there, these languages were changing fast, and men actively adopt languages of Elves when find each other and start cultural relationship. So modern (read 3rd age) mannish tongues don't contain these references to orcish. Both these remarks were absent in later version of Lhammas. Moreover tree chart of languages from this text showing their relationship, states that all languages were evolved from Valarin, which was changed in Silmarillion where tongues of elves, dwarves and men were invented by their own without contact to Valar.

The Orcs were first bred by the Dark Power of the North in
the Elder Days. It is said that they had no language of their own,
but took what they could of other tongues and perverted it to
their own liking; yet they made only brutal jargons, scarcely
sufficient even for their own needs, unless it were for curses and
abuse. And these creatures, being filled with malice, hating even
their own kind, quickly developed as many barbarous dialects as
there were groups or settlements of their race, so that their Orkish
speech was of little use to them in intercourse between different
tribes.
                                (Appendix F to The Lord of the Rings)

As the powers of evil were strong and centralized in one region in the 1st Age, it's assumed there were still less contradictions in language between orcish tribes than in 3rd age despite the quotation above.

Few words (like “golug”, Noldor elves) and names of 1st age orcs are either similar to Sindarin or raw roots from Etymologies, thus closer to Primitive Elvish. Adding the fact that first orcs were made from captured, tortured and perverted elves, we can assume that 1st Age Angband Orcish was closer to the Elvish languages than Classical Black Speech.

One may ask, why to include words into black speech from Angband Orcish, which is unrelated language. Majority of linguists consider orcish names from 1st age as Sindarin. Anthony Appleyard who used the term “Angband Orkish” (attested “golug”, “-hai”, “oghor”) counted names as Elvish except “Gothmog” (which is Elvish origin). But there will be shown some arguments on separating early Orcish from Sindarin and some inheritance of that language in Black Speech. Author is aware of some possible contradictions, so they are present too.

1. Many orcish names differ from standard Sindarin words:

Cons: these words are included in Sindarin too, with the same form in compound words, maybe there are special rules on combining words.

Pro: these compound words refer to evil creatures (like “daug”), so they may be borrowed back into Sindarin. The example of this is “gûl” (evil sorcery, necromancy) from BS “Nazgûl” from Etym. “ÑGOL” (wisdom), in contrast with “(n)gol(l)” (wise) and “angol” (deep lore, magic) derived directly from the same stem. Why many forms with “au” are considered archaic, and “o”-forms from 1st Age Orcish became a “modern” Sindarin?

Cons: some other names are clearly pure Elvish origin.

2. Some orcish words of 1st age do not have clear etymology from elvish languages like “oghor” (Drúedain) or differ more than other orcish words of that time, like “golug” and “othrod” (see table above). While Tolkien removed Orcish from the tree of languages, it is still mentioned in Appendix F of LOTR. Therefore existence of Orcish language(s) which differ(s) from Elvish is confirmed.

3. Some names and their parts occur both in 1st age an 3rd age: Gothmog (1st age Balrog, 3rd age Nazgul) and Gazmog (3rd age orc), Gorgol (1st age) and Gorbag (3rd age), Lug (1st age) and Lugbûrz, Lugdush, Lughorn (3rd age) – meaning of “lug” is different in ages but may be connected. Suffix “-hai” in names of races and nations is also found throughout the ages.

Cons: both “Gothmogs” are not orcish names, “Gazmog” was dismissed, “Gorbag” may be from “Gorb + ag”, explanation for connecting different meanings of “Lug” is just a speculation.

4. A lot of Classical Black Speech of Ring inscription and other assorted word are also borrowed from Elvish languages. Orcish names in Debased Black Speech add even more elvish roots:

Pro: Both languages borrows from Elvish heavily.

Cons: It seems newer CBS prefers to take words from more archaic Quenya and Valarin rather than from Sindarin as older 1st age “Orquin”.

Conclusion:

1. While it cannot be fully proved that names of orcs and other creatures of 1st age are not Sindarin, Tolkien explicitly stated that orcish language(s) existed in older days; therefore it was at least a perverted dialect of Sindarin;
2. Both languages have signs of belonging to Elvish group of languages;
3. Despite Angband Orcish clearly differs from Classical Black Speech, there are hints that some words have migrated into Black Speech directly or through common ancestry.