You will find Joseph and family on the 1851 census indexed as Broadley.
Lymm, Cheshire Joseph Broady Head 30 Cordwainer Lymm, Cheshire Sarah Broady Wife 38 Lymm Mary A. Broady Dau 8 Lymm Emma Broady Dau 6 Lymm Betsy Broady Dau 3 Lymm Ho/107/2163 Folio 136 Page 14
In 1841 they are indexed as Brady....but the image is very difficult to read.
1841 Lymm, Cheshire John Broady 49 Cordwainer Betty 40 Betty 20 Joseph 20 Cordwainer William 17 Cordwainer George 15 Cordwainer Thomas 13 James 10 Ann 9 Peter 1 Samuel 1 All born Lymm HO107/93 Book 19 ED10 Folio 5 Page 4 Note ages are rounded down on 1841 census for adults.
There is a birth indexed for a Betsy Broady September Q 1847 Altricham 19-15. See: http://www.freebmd.org.uk/cgi/search.plhttp://www.freebmd.org.uk/cgi/search.pl The birth certificate will give you Sarah's maiden name.
You can send for the cert on line from the General Register Office: http://www.gro.gov.uk/gro/content/http://www.gro.gov.uk/gro/content/ Safe and easy to do. I do it all the time from the USA and it takes about 3 weeks to get the cert.
There is also a marriage for a Joseph Broady in 1842. The only Sarah on the page is Sarah Cowsill and I think she may be the one who he married as she is living just about dozen houses away on the 1841 census.
I hope this is helpul Good luck in your searching. Pam B.