For detailed TV schedules and announcers by year, see my complete research on British Open TV history over at 506sports (free registration required to view).
<EDITED on 7/13/2015 to bring this up to date as of the 2015 tournament>
Chronology of British Open TV coverage
1961 - ABC had planned to provide taped coverage of the Friday 7/14 final round on Wide World of Sports the next day, but ABC scrapped those plans when the Friday action was rained out. At the time, the British Open was scheduled as a 3-day event ending with 36 holes on Friday. The 36-hole finish took place on Saturday with no US TV coverage.
1966 - The Open changed to a 4-day format with the final round on Saturday. For the first time, ABC televised the event live. The network provided 1.5 hours of coverage via Early Bird satellite of the Saturday round with McKay and Byron Nelson calling the action.
1968 - Chris Schenkel worked alongside Nelson for the telecast which covered the final 4 holes.
1969 - ABC increased to 3 hours of live coverage. This was the first time the Open was televised in color in the US.
1970 - After 4 consecutive years of live telecasts, ABC regressed to late afternoon tape delay coverage of the Saturday final round and would continue this throughout the 1970s. ABC also reduced its coverage to 1.5 hours. Dave Marr joined the ABC crew. On Sunday, ABC added tape delayed coverage of the 18-hole playoff.
1972- ABC expanded the tape delayed coverage to 2 hours and this would remain the case through 1979.
1975 - ABC added coverage of the Sunday 18-hole playoff on tape delay. ABC started using Peter Alliss from the BBC on the Open. Alliss has split time on the BBC and US TV networks on Open coverage ever since.
1980 - The Open shifted to the modern Thursday-Sunday schedule. On Saturday, ABC carried 1 hour on tape delay, but the network provided 2 hours of live coverage on Sunday. This was the first live coverage of the event on ABC since 1969.
1981 - ABC provided 1.5 hours of live coverage on Saturday for a weekend total of 3.5 hours.
1982 - ESPN provided 11 hours of coverage of the Thursday and Friday rounds. Jim Simpson and Lou Palmer were among the announcers during the early years of the ESPN coverage. ABC increased to 2 hours on Saturday. Jack Whitaker joined the ABC team.
1984 - McKay did not work the Open as ABC kept him in the US to prepare for the upcoming Los Angeles Olympics. Whitaker anchored the ABC telecasts. ABC increased the Sunday coverage to 3 hours.
1988 - Due to heavy rain, the final round was delayed until Monday. ABC added 1.5 hours of tape delayed coverage of the Monday finish.
1990 - ABC expanded to 4 hours each day on the weekend. Roger Twibell anchored the ABC coverage.
1992 - Brent Musburger took over as the ABC lead announcer with Steve Melnyk as lead analyst.
1993 - Peter Jacobsen worked as lead analyst for ABC, but Melnyk would resume this role again the following year.
1997 - Mike Tirico took over as the ABC lead announcer.
1998 - ABC installed Curtis Strange as its lead analyst.
1999 - ABC moved the Saturday coverage to ESPN around 11:30am so that ABC News could follow the search for the missing plane of John F. Kennedy Jr.
2000 - ABC increased to 4.5 hours on each weekend day.
2001 - ABC expanded to 5.5 hours on both Saturday and Sunday and would continue this schedule through 2009.
2003 - TNT took over the Thursday and Friday rounds. Ernie Johnson and Bobby Clampett anchored the TNT coverage which also used some of the ABC announcers. TNT also started providing 2 hours of early morning coverage on both weekend days prior to the ABC air-time.
2004 - ABC installed Ian Baker-Finch and Nick Faldo as co-#1 analysts.
2005 - Paul Azinger joined Faldo as ABC co-lead analyst. Faldo made the cut in the event and joined the ABC coverage after finishing play.
2006 - Tirico filled in on TNT for Johnson who was undergoing chemotherapy.
2009 - Watson had again been scheduled to join the ABC weekend telecasts. However, he never made it to the broadcast booth as he not only made the cut, but nearly won the tournament.
2014 - ESPN shifted its Saturday coverage to start at 5am ET after the tournament moved tee times earlier due to an ominous weather forecast.
Total scheduled US TV time for the British Open by year
Note: I am not including added coverage (such as playoffs) in these numbers, but I did mention these cases in the above chronology. I am also not including taped highlight shows, but am including original tape delay coverage. The hours represent live coverage except where noted.
1962-1965: at most 1.5 hours (ABC) - covered on tape as part of Wide World of Sports
1966-1968: 1.5 hours (ABC)
1969: 3 hours (ABC)
1970-1971: 1.5 hours on tape delay (ABC)
1972-1979: 2 hours on tape delay (ABC)
1980: 3 hours total with 1 on tape delay and 2 live (ABC)
1981: 3.5 hours (ABC)
1982: 15 hours (11 on ESPN, 4 on ABC)
1983 14 hours (10 on ESPN, 4 on ABC)
1984-1989: 15 hours (10 on ESPN, 5 on ABC)
1990-1999: 18 hours (10 on ESPN, 8 on ABC)
2000: 21 hours (12 on ESPN, 9 on ABC)
2001-2002: 23 hours (12 on ESPN, 11 on ABC)
2003-2004: 30 hours (19 on TNT, 11 on ABC)
2005-2006: 31 hours (20 on TNT, 11 on ABC)
2007-2009: 30 hours (19 on TNT, 11 on ABC)
2010-2011: 37 hours (ESPN)
2012: 36 hours (ESPN)
2013-2015: 37 hours (ESPN)