Question: What was a mountain building period that ended 440 million years ago, that durring this time a unit of interbedded greywackes and shales that outcrops in eastern New York State were deposited in a deep marine setting in a foreland basin during this time? 

Context paragraphs:
Geology of Surrey: Geology of Surrey.  Towards the beginning of the Cretaceous age (146–66 million years ago) Surrey alternated between a fresh-to-brackish water embayment depositing Hastings Beds and Weald Clay, comprising shales and mudstones that are often finely banded.  Offshore muds (now shales and mudstones) of the Atherfield Clay were deposited followed by shallow marine sands of the Hythe, Sandgate and Folkestone Beds.  Where not eroded to lower heights, there is then a marine layer of the sands of the Hythe Beds topped by chert seen on today's remaining Greensand Hills.  Instead of the mudstone and sandstone-producing three beds mentioned before Hythe Beds, west of Dorking the marine Bargate Beds made of calcareous (chalk and limestone-rich) sandstone were deposited.  The Folkestone Beds contain phosphatic and iron-rich nodules, which locally yield a rich fossil fauna of marine shells.

Himalayan foreland basin: The Himalayan foreland basin is an active collisional foreland basin system that developed adjacent to the Himalayan mountain belt in response to the weight of crustal thickening when the Indian Plate collided and subducted under the Eurasian plate.  With a width of 450 km and 2,000 km long, the foreland basin span to five countries which include: India, Nepal, Pakistan, Bhutan, and Bangladesh.  Sedimentary rock records gathered on the greater Himalayan area surrounding India and Nepal traces back to the onset development of the foreland basin which started at the Paleogene period around 45-50 Ma to the time where India and Asia collided.  A debate has arisen over whether the basin exhibits the concept of progressive suturing (diachroneity), which means a lag in time exists between parts of the basin; it was inferred the development of the western part of basin started at the Paleocene period around 57-54 Ma with the central part having 2 Ma in difference, and gets younger as it progresses towards the eastern side.  The various stratigraphic succession of the basin is important as it preserves the evidence of the India-Eurasia collision as well as relating it to the Himalayan Orogenesis.

Clinton Group: The Clinton Group (also referred to as the Clinton Formation or the Clinton Shale) is a mapped unit of sedimentary rock found throughout eastern North America.  The interval was first defined by the geologist Lardner Vanuxem, who derived the name from the village of Clinton in Oneida County, New York where several well exposed outcrops of these strata can be found.  The Clinton Group and its lateral equivalents extend throughout much of the Appalachian Foreland Basin, a major structural and depositional province extending from New York to Alabama.  The term has been employed in Kentucky, Maryland, Michigan, New York, Ohio, Pennsylvania, Tennessee, Virginia, and West Virginia, though in many of these areas the same interval is referred to as the Rose Hill, Rockwood, or Red Mountain Formations.  Historically the term "Clinton" has also been assigned to several lower Silurian stratigraphic units in Ohio and Kentucky which are now known to be significantly older than the Clinton Group as it was originally defined.  Many parts of this succession are richly fossiliferous, making the Clinton Group an important record of marine life during the early Silurian.  Several economically valuable rock-types are found within this interval, though it is perhaps best known as a significant source of iron ore

Ausia (animal): Ausia fenestrata is a curious Ediacaran period (630 – 542 million years ago) fossil represented by only one specimen 5 cm long from the Nama Group, a Vendian to Cambrian group of stratigraphic sequences deposited in the Nama foreland basin in central and southern Namibia.  It has similarity to "Burykhia" from Ediacaran (Vendian) siliciclastic sediments exposed on the Syuzma River of northern Russia.

Flysch: Flysch is a sequence of sedimentary rocks that is deposited in a deep marine facies in the foreland basin of a developing orogen.  Flysch is typically deposited during an early stage of the orogenesis.  When the orogen evolves, the foreland basin becomes shallower and molasse is deposited on top of the flysch.  It is therefore called a syn-orogenic sediment (deposited contemporaneously with mountain building).

Cobequid fault: The Cobequid Fault is a fault that is the extension of the Chedabucto Fault.  This fault runs from Truro to Cape Chignecto south of the Cobequid Hills.  Both the Cobequid fault and the Chedabucto Fault form the Minas Fault zone splitting Nova Scotia into the Avalon Zone and the Meguma Zone.  Movement on the Cobequid Fault started before 400 million years ago and end around 40 million years ago.  Between that time around 350 million years violent volcanic eruptions at Spicers Cove north Cape Chignecto Provincial Park and 50 million years after grey sandstone rich in fossil plants were deposited.  At Five Islands Provincial Park there are red rocks formed by the large accumulation of mud, sand and gravel around 210 million years ago.

Taconic orogeny: The Taconic orogeny was a mountain building period that ended 440 million years ago and affected most of modern-day New England.  A great mountain chain formed from eastern Canada down through what is now the Piedmont off the East coast of the United States.  As the mountain chain eroded in the Silurian and Devonian periods, sediments from the mountain chain spread throughout the present-day Appalachians and midcontinental North America.

Naukluft Mountains: The Naukluft Mountains are a nappe complex: a stack of rock units which have been transported toward the southeast on a basal thrust fault.  The first geological maps and cross sections of the mountain range were made in the 1930s by Henno Martin and Hermann Korn.  The rocks are interbedded sediments, dominated by dolostones, quartzites and shales, deposited in a shallow marine environment in the hinterland of the Damara Orogeny between 700-500 million years ago.  The sediments are equivalent in age to the Otavi Mountainlands sediments where evidence for snowball earth climates was described by Paul F. Hoffman.  The same lithologies are present in the Naukluft Mountains, but the sediments are folded and faulted so the original stratigraphy is deformed and inverted.

Laramide orogeny: The Laramide orogeny was a period of mountain building in western North America, which started in the Late Cretaceous, 70 to 80 million years ago, and ended 35 to 55 million years ago.  The exact duration and ages of beginning and end of the orogeny are in dispute.  The Laramide orogeny occurred in a series of pulses, with quiescent phases intervening.  The major feature that was created by this orogeny was deep-seated, thick-skinned deformation, with evidence of this orogeny found from Canada to northern Mexico, with the easternmost extent of the mountain-building represented by the Black Hills of South Dakota.  The phenomenon is named for the Laramie Mountains of eastern Wyoming.  The Laramide orogeny is sometimes confused with the Sevier orogeny, which partially overlapped in time and space.

Austin Glen Member: The Austin Glen Member of the Normanskill Formation is an upper Middle Ordovician unit of interbedded greywackes and shales that outcrops in eastern New York State.  It was deposited in a deep marine setting in a foreland basin during the Taconic orogeny.  Its sediment source was mainly the erosion of preexisting sedimentary rocks.  Graptolite fossils place it in the stratigraphic zones of "Nematograptus gracilis" and "Climacograptus bicornis", but its age could be Llandeilo or Trentonian (earliest to latest Darriwilian,  465 ).

Supporting paragraphs: Austin Glen Member Taconic orogeny
Answer: Taconic orogeny