In writing, as 'Lord Mayor', perhaps surprisingly. The title Lady Mayoress is reserved only for the wife of a Lord Mayor. You should address a female Lord Mayor as "Madam Mayor" in conversation. It is considered impolite, certainly in public speaking, to address her just as 'mrs. Smith'.
In writing an envelope, you use "Mrs. J.B. Smith, Lord Mayor of Portsmouth".
As Mayor, Madame Mayor, or by name.
"The Right Honourable Lord Mayor"
Dear Madam Mayor
Whether male or female it would be: Mayor Jane Doe or Mayor Doe (as example.)
Lucy Turnbull.
Dame Mary Donaldson - 1983
In Great Britain, a Mayor is addressed as 'Your Worship'. In the United States, you address him as Mr. Mayor, Your Honor or simply as Sir. If the mayor is female, you address her as Madam Mayor or as Ma'am. The mayor can be referred to as the "Honorable John Doe".
There is no lord and lady of Dublin. Dublin has a Lord Mayor, who can be male or female and who holds a one year term, from June each year. In June 2014, Christy Burke was elected the Lord Mayor of Dublin and will hold the position until June 2015.
Lord Mayor is the title of the Mayor of a major city. Great Britain is a country and as such doesn't have a Lord Mayor.
The Wife of a Lord Mayor in the United Kingdom is properly and officially refereed to as a Lady Mayoress, citing the suffix -ess as applied for feminine titles in English. Many former British Colonies follow suit, such as in Australia. Similar to the designation "First Lady", the Lady Mayoress has a symbolic nature towards events and the nation in general towards the area she represents. When the Lord Mayor is female, she usually is not designated as the Lady Mayoress, but takes the title of Lord Mayor. There have been cases, however, of a female as Lord Mayor and her mother as Lord Mayoress, such as in Brisbane,UK. In America, no such designation is official, but it is still a recognized term for the wife or a lady holding the position of mayor, although is not very popular as an American designation.
Mrs John Governor
Yes, "Lord Mayor" should be capitalized because it is a title referring to the mayor of a city or town.