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I will check on the Winkles if I get a chance to call on the archives again (possibly this weekend if they are open)

some of the Liverpool records are also missing on line but a visit to Liverpool is a bit more difficult for me

Yes, Liverpool is only in snippets, I wouldn't have even asked there.

Speaking of missing records, the worst for my tree, by far, is Gloucester and the sheer drought of records available for it. It seems like very little is available, and I can do almost nothing for my line in there.

Yes if you give me the dates I will see what I can do.
Co-incidentally I have both Sawyers and Winkles in my family tree.
Samuel Winkle (1887 - 1947) married my great grand aunt Minnie Bourne in 1909. His father was Frank Winkle (1861) and he was one of seven children.
and
Samuel Sawyer (1823-1895) married my 4th great grand aunt Elizabeth Bourne in 1843. His father was John Sawyer (1789-1861)
All lived in Burslem

Not sure how Samuel Sawyer links  up with the rest of my Sawyers (if he does at all).  Ralph Sawyer only had one sibling, Hugh, who doesn't seem to have survived though I've never found a burial record.  Ralph and Hugh's father was another Ralph Sawyer, the only child that I have been able to positively link to John Sawyer (buried Burslem 1806) and his wife Mary (____) (buried Burslem 1800).  There are several John Sawyer and Mary marriages but none very close to Burslem, so I am unsure which if any that John and Mary couple are.

My Joseph Winkle is probably the son of Joseph Winkle and Hannah Gibson, but I am still working on it.

Here are the Sawyer dates:

Henry Sawyer was born around 1823 (source: given age at burial, age in censuses)
Levi Sawyer was born around 1825 (source: census of 1841, US federal census of 1860)
Edwin Sawyer was born around 1826 or 1827 (source: census of 1841, US federal census of 1860)
Harriet Sawyer was born around early 1841 (called one month old in 1841 census, claims a Feb 1841 birth date in 1900 US federal census)

Winkle. These dates come from FamilySearch and are not on freereg.:

19 Apr 1812 Thomas at Hanley
28 Nov 1813 William at Hanley
8 Oct 1815 Elizabeth at Hanley

I have been able to sort out the Winkle children who were baptized at Burslem, because it helpfully gives the occupation of the father.  Also the abode of the parents would help, as my Joseph Winkle's abode is usually given as Cobridge, while the other couple is usually given as Scotia or Burslem.

If Ralph and Mary were potters from Staffordshire living in Liverpool then it is quite likely that they were working at the Herculaneum factory at Toxteth Park which ran from 1796 - 1840. Almost the entire workforce were employed from the Staffordshire potteries, travelling to Liverpool by barge. When the factory closed in 1840 many of them returned to Staffordshire.
Details of the pottery are covered in "The Herculaneum Pottery -Liverpool's forgotten glory" by Peter Hyland
Extracts from the Lancashire parishes can be found here : http://www.lan-opc.org.uk/Search/indexp.html
I have missing baptisms from ancestors that worked there. For though there was a chapel and many baptisims I believe that children born on the journey between Liverpool and Burslem may have been baptised along the way or else not baptised at all.

Thank you for the information.  That is definitely possible, but I now strongly believe that the missing Sawyer baptisms may have happened at Hanley parish.  There is a huge gap in coverage Findmypast/Freereg for that parish between 1804 and about 1841.  A few of the baptisms after that are on familysearch, but not many.   Not just Levi's baptism is missing, but the other two middle children - Henry (the only child who came back and stayed in England) and Edwin, as well as the last child Harriet.

I am looking for information on a Levi (in some trees he is called Levi Joseph Sawyer, but I don't know where the Joseph comes from) Sawyer, supposed to have been born in Burslem Staffordshire on 17 Mar 1825. I looked through what was available for this parish on familysearch, and while it showed some of his reputed siblings, nothing for him.

I found this old thread for what is supposed to be the same family:

http://www.rootschat.com/forum/index.php?topic=561525.0

There is a book mentioned in that thread, The Richard and Harriet Hopkins family, Empire Prairie pioneers, which covers what is supposed to be the same family, but Levi is only mentioned in passing. 

There are a couple of records on Ancestry.com, "Seventy Quorum Membership, 1835–1846", and "Membership of The Church of Jesus Christ of Latter-day Saints, 1830-1848", which give Levi Sawyer and 1839 LDS baptism date, and also give his parents as Ralph Sawyer and Mary Winkle, but I cannot find the works that they base their information on (i.e. contemporary record of this parentage, although I do believe it is 99% likely).

Levi does seems to fit with this Wisconsin Sawyer family in the above book, but they are supposed to have immigrated in the early 1840s, so I don't know how Levi got baptized LDS in 1839, unless it was in England, but the book doesn't mention any Mormon connection.  However, in looking at the history of LDS missionary work in England, the Staffordshire potters seem to have been early targets for conversion. 

In the 1841 Census, in Burslem, there are a John and Joseph Sawyer living together.  John's age roughly corresponds with John, son of Ralph, baptized in 1820, and the 'Joseph' is said to be about 15 in 1841, which is close to Levi's stated birth date of 1825.  However, their occupations both seem to be called mining (tough to read), and not potters, and John appears in the household with his father Ralph in the 1841 census.

I have seen some mention online that he married an Elizabeth Bell in 1845 in Nauvoo IL, but I couldn't find a record of it.  He did marry Rose Ann Ball in WI in 1851, and she appears with him in the 1855 Wisconsin State Census and later the Federal 1860 Census in Utah. I cannot find him in 1870 or 1880, but he does appear, as a widower, on the New Mexico 1885 Territorial census with his family, including my great-great-grandmother Rosa Belle, as well as his oldest son named Ralph - however, online trees have this son Ralph being born in 1851, and dying in 1856 or so, but the census clearly calls this Ralph Levi's son.  His age is also off from the 1851 Ralph, and this young man does not appear in the 1860 census with Levi.

The Ralph Sawyer/Mary Winkle family also seems to have lived in Liverpool for a time.