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So, I guess this is her death - which means the names are wrong way round:

Mary Teresa Burgess
Death Date: 1 Jan 1905
Death Place: Queensland
Father's Name: Michael Flannigan
Mother's Name: Annie Collins
Registration Year: 1905
Registration Place: Queensland
Registration number: 000383
Page Number: 3343

Puts a slightly different slant on searches.

The surnames are, of course, all Irish - perhaps you should try

https://familysearch.org/

Have you tried searching 'arrivals' in Australia?

Emigration records are not on line for years before 1890

Was her father named on her marriage cert?

Is there any 'occupation' info for him?

Thanks. I have looked up the first marriage on freebmd and cannot find it although one possibility is a Theresa Bridget Flanagan who married Michael Sullivan in 1882 in Queensland. This doesn't seem close enough for me to pay for the certificate just yet, although I realise I may have to do that if I can't find any more leads. I'm afraid I have no information on when she moved to Australia and whether she married before or after her arrival. This is where the difficulty lies. Thanks for your interest.

Look for the first marriage on freebmd

When did she move to Australia?

What was her surname at the time?

I would appreciate help finding the ancestors of Teresa Mary Flannigan. The facts I know are as folllows:
She married Isaac Burgess Hudson (also known as Isaac Hudson Burgess) in Queensland, Australia in 1900 and died in Queensland in 1905. Her death certificate states she was 53 years old (so born around 1852) and was from London, Middlesex, England. Her parents are named on the death certificate was Annie Collins and Michael Flannigan. She had spouse prior to Isaac whose surname was Sullivan. The death certificate and marriage certificate tell me no more information about her first husband (it just says "unknown").
I am trying to find her ancestors but I find myself stuck where to go from here. Any help is appreciated.