How big is the UK financial services industry? How big is the global
financial services industry?

Might want to define Financial Services industry
for the researchers, like Banking, Finacial Advisors, etc.
Surprised no one gave you an answer yet.

Further to the last comment. If you specify your interest you can get
the information easily. Banking and Insurance are broadly different
categories. If you are interested in only the high level gross size
the International Monetary Fund publishes a monthly series called the
International Financial Statistics. That has every IMF member country
data. Details probably will come only with a higher price.

What I mean by the financial services industry is primarily banks,
brokers, insurance companies and fund managers. What are the chances
of my getting an answer to this? How do I access the IMF data
mentioned?

A few brief figures for the UK:
The UK is responsible for 20% cross-border lending worldwide.
Financial services produce 5.1% of UK GDP, employ c.1 million people,
and have net overseas earnings of £14bn in 2001. The UK insurance
sector is the largest in Europe, employs 340,000 people, accounts for
£8 billion of annual UK overseas earnings,  More than 20% of stock
market investments are related to insurance. Each day, £225 million
goes out in pensions and life insurance benefits as well as £41
million for general insurance claims.
http://www.cssa.co.uk/sectors/market_financial_services.asp

Thank you!