A small wooden building “then considered an old one, and erected in another part of the town [that] was purchased and removed on wheels to the allotment in Frederick-street, where it was converted into a chapel”.
1839 Opened
1842 Replaced by St John’s Square Chapel
1885 Demolished.

From the 1886 Assessement Roll for Launceston. The land marked replaces 3 occupied houses in the 1885 roll, which makes it likely to be the site of a wooden tenement proximal to the Primitive Methodist Chapel demolished in the previous few months. That site would be either next door or one building down from the Primitive Methodist Chapel (depending on the location of its associated house). Thus, it would be approximately here:

Towards the close of 1838 the Rev. John West arrived in Tasmania, having been sent out with the Rev. Alexander Morison, now of Melbourne, by the Colonial Missionary Society to carry on mission work in the northern districts of this island. Whilst thus engaged Mr. West occasionally preached in Launceston, where he gradually gathered around him a band of earnest fellow workers, who ware charmed by his gentle winning manner and his deep thoughtful teaching. On 12th June, 1840, the church fellowship was formed, consisting of eight male and ten female members, one of the latter being the sole survivor to-day. The infant church met for worship in the Infant School whilst a small wooden chapel was being prepared in Frederick-street.
Launceston Examiner, 20 October 1885

Infant School

Launceston Advertiser, 29 August 1839

On Sunday last (September 1) the Independent Chapel, lately erected in Frederick-street, Launceston, was opened for Divine Service. The Sermons on the occasion were preached by the Rev. H Dowling, Baptist Missionary, and the Rev. John West, of the Colonial Missionary Society. We understand that the congregational church connected with this place of worship, stands in union with, and has been recognised by, the Van Diemen’s Land Congregational Union, at Hobart Town.
Launceston Advertiser, 12 September 1839

A MEETING was held on Friday last amongst the members of the Independent Chapel, Frederick-street, for the purpose of taking steps to erect a new chapel upon an allotment of ground adjoining the residence of Mr. Sams. It was unanimously agreed that the present chapel is far too small for the rapidly-encreasing congregation, and that another more commodious should be immediately commenced. The almost incredible sum of £580 was subscribed by the members and congregation present. We have often had occasion to extol the spirit of liberality which is so universally displayed amongst our little community, in all matters either of religion or education. There is not a place in the world where, in comparison with the number of inhabitants, so many places of worship could be found. In addition to this intended chapel, there is a handsome church being erected for the Catholics, on the square at the foot of the Cataract Hill. The Wesleyan Chapel, Patterson-street, the Independent Chapel, Tamar-street, and the Chapel in Frederick-street, the Baptist Chapel in York-street, and the Catholic Chapel in Cameron-street, have all been built within the last four or five years, and principally by public subscription.
Launceston Advertiser, 15 April 1841

Mr West arrived here some five and thirty years ago, and after a brief missionary career in the Northern rural districts, he was invited to form a second Congregational Church in Launceston. A wooden building, purchased and removed to a site in Frederick-street, opposite the residence of Mr James Henry, was converted into a chapel, and here for a few years he officiated as pastor with great acceptance ; until the congregation increasing in numbers rendered it necessary to procure a larger chapel, and the one fronting on Prince’s Square was erected.
The Examiner, 30 December 1873

Launceston Examiner, 6 August 1842

Launceston Examiner, 21 December 1842

A LOCAL ANTIQUITY.-Amongst the old buildings recently ordered, by the Municipal Council to be demolished as being unfit for habitations is one that has played a somewhat conspicuous part in this good town of Launceston. We allude to the old building, the gable end of which fronts on the north side of Frederick-street, adjoining the primitive Methodist Chapel, and about midway between Charles and Wellington streets. When a second Congregational Church was formed in Launceston, about the year 1839 or 1840, the late Rev. John West being the pastor, the building in question–then considered an old one, and erected in another part of the town–was purchased and removed on wheels to the allotment in Frederick-street, where it was converted into a chapel, and was used for public worship for several years. When the present brick church, fronting the Square, and known as Prince’s Square Church, was erected, the former edifice was converted into dwelling houses, and it has continued to do duty in that shape for the last forty years. “To what base uses we may return, Horatio!” It is not a little singular that just when the edict of demolition has gone forth against the “old original,” a third and still larger church–its offspring, so to speak–is on the eve of being opened, in which will be carried on the work inaugurated so many years ago in this humble edifice.
Launceston Examiner, 23 May 1885

APPEAL UNDER POLICE ACT AMENDMENT, 45 VIC , NO. 22, SECTION 6
This was an appeal by the trustee in Swan’s estate against the notice of the Launceston Corporation, under the Police Act Amendment.
45 Vic, No. 22, section 6. The tenement of which notice was given to be pulled down is an old wooden building in Frederick-street, between Charles and Wellington streets.
[continued]
The Tasmanian, 11 July 1885