This subject is touched on briefly in The Book of the Rat, an independent work drawn up for the author's Skaven-themed WFRP campaign. This seems to predate The Children of the Horned Rat, an officially published supplement for WHFRP 2nd Edition.
Others have attempted to fill in the gaps. In my investigations I cam across the web-pages of the Belgarth Medieval Combat Society, who have run LARPs taking place in the Warhammer world and included the Skaven. They makes extensive use of the information provided by TBotR, and expand upon it. The Belgarthwiki contains this information regarding the Skaven Alphabet:
"Few humans have ever ventured into Skavenblight, and fewer still have ventured out. For many decades, despite the appearance of skaven runic symbols on battlefields across the old world, it was believed that the Skaven lacked a written language of their own. In support of this theory, it was pointed out that Skaven battle standards are often written in the common tongue, bearing such epitaphs as "we shall inherit" and "wither and decay." However, we now know that the Skaven use the common tongue solely for their opponent's benefit - in an attempt to break enemy morale. Only recently, through the interrogation of human cultists of the Horned Rat, and careful study of a few captured tattered pages of the "Liber Boubonicus" have the mysteries of the Skaven written language begun to unravel.
The Skaven alphabet contains 19 letters. There is no hard and fast rule concerning whether material is written from left to right, right to left or up and down.
Each of the 19 letters takes the form of a runic symbol. These symbols, when appearing alone, have a distinct meaning within Skaven society. In addition, however, the word or phrase associated with each symbol also represents a certain sound in the Skaven alphabet. For example, the skaven symbol for the word "warpstone" may appear alone, and often does. However when it appears with a group of other symbols it may also represent the "Wa" sound, or the letter "W".
In addition to their 19 letters, the Skaven alphabet contains 7 vowels. The vowels are often omitted in informal writing, as the Skaven vocabulary appears to be so limited as to have little need of them. Thus, a word like "water" will usually appear using the runic symbols for "W", "T" and "R" only.
Finally, the Skaven use a single punctuation symbol in their writings: the double scratch mark, or "//" symbol. When this symbol appears after a word, the word is always pronounced in a stuttering fashion, such as "Die-die" or "Hate-hate". No other punctuation is known to exist in the Skaven written language. It is believed that the question mark is not used because written requests are of no importance in Skaven society. No Skaven would ever respond to a request such as "where is the warpstone kept." Such information would only be given in person, either to a more powerful Skaven, or extracted by his assassins."
An interesting summary, this is further explored here:
"While Skaven have a native language called Queekish, they fluently speak the Manthing tongue so as to better strike fear into the hearts of their enemies. The Skaven have made a few improvements to the Manthing language showing their superiority over their weak and stupid enemies. These changes are as follows.
- Skaven chatter a lot. In the Manthing tongue they will, in their excitement, repeat words for emphasis. Ex. Die-Die Manthing!! Run-Run!! Skuttle Quick-Quick!!!
- Skaven view themselves as superior to all other races. They refer to civilized races as Manthing, Dwarfthing, Elfthing, etc.
- Skaven in their political cesspool have mastered flattery. When asking a favor, in fear of being punished, or other dealings where they may need your cooperation they will shower you with compliments such as "Most prestigious of prestigious Lords" or "Greatest of vile vermin."
- Skaven have mastered mud-slinging. When enticing someone to work against another Skaven or another Clan they will detract from their intended target with phrases like "That backstabbing coward of Clan Skryre, the Horned One chew-chew his intestines!!"
- Few female Skaven escape the breeding warrens of Skavenblight. Skaven refer to all females as breeders. They have no other word for the female gender.
- When Skaven are excited they tend to chatter uncontrollably: Neek-Neek.
- When Skaven are afraid they make high pitched whining sounds: Eeek-Eeek!!!
- Although not a part of the spoken language, Skaven tend to touch what ever they are talking to.
EDIT: I've removed the link to Bill Lippman's article; a quick google and anyone can find it if they are interested.
The situation is not much improved when we look at the consonants. Let's ignore for a second that several of the runes are far too complex in nature to functionally be used as letters in a useful writing system. Instead, for example look at the symbol for "B". The explanation is that the symbol used is a pictogram representing a "bell" and hence the symbol used for "b". lolwhat? That would only make sense if the word for "bell" in Queekish was also, unbelievably, bell. Almost every letter is an example of this in some manner.
Hence my description of this as a cipher for English. This alphabet only makes sense in the context of translating modern English words into runes, like a code. Granted, I can understand the utility of this for the purposes of gaming, but this also implies that the common tongue in the Oldhammer world is actually modern English, whereas I accept the English used in Warhammer fantasy as a direct translation of what is likely a proto-Germanic tongue, much as the language used in the Lord of the Rings is assumed by Tolkien as a direct translation of an ancient British dialect. It in no way realistically represents the alphabet of a unique language developed by The Skaven themselves.
And all available evidence suggests that the Skaven would have developed their own tongue. In fact, linguistically, the Skaven are shown to be exceptionally advanced. I recently read the surprisingly excellent Rise of the Horned Rat novel, detailing a major Skaven campaign during the End Times to eradicate the Dwarves. In it, Queek, who advances to the ancient Skaven age of ten years old (!), despite being specifically depicted as a, shall we say, less-intellectually-inclined, still displays an uncanny mastery of a variety of tongues. Queek even specifically uses this knowledge to torment his opponents, choosing to taunt Dwarven opponents the sacred language of their ancestors. As its inconceivable that any Dwarven tutor would have schooled Queek in this tongue, this suggests he was able to pick it up naturally just by hearing it spoken, either in battle or by captives.
All this suggests that the native language of the Skaven is a manifold and nuanced tongue. Its not my intention at this point to create a fully-functioning artificial language, an endeavour beyond the scope of this blog or my current available free time, but at the very least a runic alphabet that suggests the depth and complexity of Queekish, and perhaps a basic grammar to provide a basic structure for the language.