Which brewery company owned the bowling greens at Mortlake, London
formerly used for holding the National Championships (outdoors, flat
green bowls)

Hello goldenbowls,

The bowling green belonged to Watneys.  

Here is an excerpt from an account about a Fred Deely of Doddington,
written by Dorothy E Clarke:

“There were two bowling clubs in Deddington. "About four of us
youngsters, and the others were shopkeepers 'and that', they called us
the 'lower fives' (Castle Bowls)….   That was in 1926 and Fred's still
playing today. [1994 CR] ''I'm reckoned to be one of the oldest
bowlers in the County. I pulled a muscle about two years ago, and I
get a twinge when its wet."…  Later, in 1969, he went to London to
play at Mortlake, outside Watney's Brewery. He was in the last eight
and lost by one shot.”
http://www.deddington.org.uk/history/freddeely2.html

The bowling greens are still in existence.  If you go to
http://www.barneseagles.com/ba_grounds.htm and scroll about halfway
down, you will reach Watney's Sports Ground, Lower Richmond Road,
Mortlake, London SW14   Select “Click here for a Map of Watney's
Sports Ground”
I’m giving that reference because the URL is very long, so I’m not
sure it will work, but here it is:
http://www.multimap.com/map/browse.cgi?X=520500&Y=176000&width=500&height=300&client=public&gride=520536&gridn=175957&srec=0&coordsys=gb&addr1=&addr2=&addr3=&pc=SW147ET&advanced=&scale=10000&left.x=8&left.y=3
There are two bowling greens marked, just below Clifford Avenue.  I
note that the brewery is now indicated as belonging to Courage. It is
actually the place where Budweiser is brewed for the UK market.

“The Middleton Cup is the English inter-county team championships held
in lawn bowls at Mortlake, London. The event started in 1911 and was
originally played for the John Bull Cup, but this was replaced in 1922
by the Middleton cup.”
http://www.probertencyclopaedia.com/O3.HTM 

search strategy: bowling Mortlake brewery

Adequate answer, confirmed my own research