Hi
The surname looks to be Brumwell not Bru(m)mell
http://familytreemaker.genealogy.com/users/s/t/u/Marc-Stuart/WEBSITE-0001/UHP-0415.htmlJANE BRUMWELL OR GORDON 
Birth:  15 APR 1807    
Christening:  07 MAY 1807   St Nicholas Parish Rec And Nonconf, Newcastle Upon Tyne, Northumberland
Father:  WILLIAM BRUMWELL  
Mother:  ELIZABETH GORDON  
ANNE BRUMWELL OR GORDON
Birth:   Christening:  26 DEC 1809   St Nicholas Parish Rec And Nonconf, Newcastle Upon Tyne, Northumberland 
Father:  WILLIAM BRUMWELL  
Mother:  ELIZABETH GORDON  
WILLIAM BRUMWELL  
ELIZABETH GORDON   
Marriage:  03 JUL 1810   Saint John, Newcastle Upon Tyne, Northumberland 
WILLIAM BRUMWELL 
Christening:  05 SEP 1813   Saint Nicholas, Newcastle Upon Tyne, Northumberland
Father:  WILLIAM BRUMWELL  
Mother:  ELIZABETH   
JOHN BRUMWELL 
Christening:  31 DEC 1815   Saint Nicholas, Newcastle Upon Tyne, Northumberland
Father:  WILLIAM BRUMWELL 
Mother:  ELIZABETH  
Another tree on the internet gives William Brumwell bapt 1754 Alston, died 1815, the son of John Brumwell 1721-1792 and Isabella 1726-1803 marrying to Susanna Wilkinson in 1783 and having a daughter Isabella in 1783
William Brumwell  
Susanna Wilkinson   
Marriage:  23 OCT 1783   Wisbech,St Peter, , Cambridge 
ISABELLA BRUMWELL 
Birth:  15 JUL 1784    
Christening:  08 SEP 1784   St Nicholas Parish Rec And Nonconf, Newcastle Upon Tyne, Northumberland
 Father:  WILLIAM BRUMWELL   
Mother:  SUSANNAH   
Not to be confused with this William Brumwell on someone else's tree on the internet baptised according to the tree in Alston in 1752 (other trees 1764) to George 1719-1804 and Euphemia M Leeke 1719-1768
UPHAN BRUMWELL (Euphemia)  
Christening:  23 FEB 1800   Hexham, Northumberland
Father:  WILLIAM BRUMWELL  
GEORGE BRUMWELL 
Birth:  22 SEP 1801    
Christening:  25 JUL 1802   St Nicholas Parish Rec And Nonconf, Newcastle Upon Tyne, Northumberland 
Father:  WILLIAM BRUMWELL   
Mother:  ANNE ENGLISH  
WILLIAM BRUMWELL  
Birth:  11 AUG 1803    
Christening:  12 AUG 1804   St Nicholas Parish Rec And Nonconf, Newcastle Upon Tyne, Northumberland 
Father:  WILLIAM BRUMWELL 
Mother:  ANNE ENGLISH 
THOMAS BROMWELL 
Birth:  04 SEP 1807    
Christening:  11 OCT 1807   St Nicholas Parish Rec And Nonconf, Newcastle Upon Tyne, NorthumberlandFather:  WILLIAM BROMWELL 
Mother:  ANNE ENGLISH       
MARY ANN BRUMWELL 
Birth:  13 OCT 1812    
Christening:  08 NOV 1812   St Andrews Brunswick Pl Formerly Orphan Hse-Wesl Mnew, Newcastle Upon Tyne, Northumberland
Father:  WILLIAM BRUMWELL  
Mother:  ANN  
WILLIAM BRUMWELL 
ANN ENGLISH   
Marriage:  12 MAY 1799   Hexham, Northumberland 
Ann English's dates according to the tree 1771-1844
These bits from the internet
'Mr. Daglish was the pupil and successor of the late William Brumwell, surgeon, a man of singular simplicity and benevolence of character. Mr. D. in speaking of his respected master, says he was "a constant friend to, and warm advocate for vaccination; and who, for a period of nearly forty years, practised in Newcastle with success and credit; whose urbanity and kindness obtained for him the esteem of a numerous and respectable circle of friends, the affection of his servants, and the gratitude of the poor.'
successor certainly by 1825
http://www.british-history.ac.uk/report.aspx?compid=43378'The trial at large of Sir Matthew White Ridley, bart., M.P., for and one of the aldermen of Newcastle upon Tyne for criminal conversation with the wife of Mr. William Brumwell, surgeon, Newcastle: before Lord Kenyon, at Guildhall, March 4th, 1793 : with the pleadings of councel at full length (Mr. Erskine for the plaintiff, Mr. Law for the defendant.) / taken in short-hand by Mr. E. Hodgson.''In fact he got himself into considerable trouble in this respect in 1793, when he was sued by a Dr. Bromel for 'Criminal Conversation' with Mrs. Bromel on the staircase of their house in Newcastle. Sir Matthew, who had been caught in 'flagrante delicto' by Mrs. Bromel's maid, was found guilty by a special jury, and ordered to pay £400 damages. This little incident did not, however, affect his public career, although he was Mayor of Newcastle and Member of Parliament at the time [27/10]. Perhaps the general public were of the same opinion as Sir Matthew's counsel who maintained, in mitigation, that Mrs. Bromel, although admittedly very beautiful, was not morally above reproach. Moreover, her husband had shown such a lamentable complacency in her affairs, that he suspected that they had conspired to rob this great but susceptible gentleman - not only of his money but of his peace and quiet. Any way, Sir Matthew continued to enjoy the confidence and affection of his constituents as well as his wife.'http://www.amazon.co.uk/Matthew-Criminal-Conversation-William-Brumwell/dp/1170003206Monthly magazine and British Register 1816
Died Mr William Brumwell of the Sandhill much respected 61
Burial 31st December 1815 St Nicholas Newcastle aged 61
There are other William Brumwells burials in Newcastle
Regards
Valda