The truth is that nobody knows for sure. There is a tendency to overstate the influence of union executives in "delivering" swathes of voters, partly a hangover from the days of the trade union bloc vote, so that the 1983 leadership contest was effectively in the bag for Neil Kinnock by teatime on the day the election was announced.
Fortunately, party democracy has come a long way - and candidates need to appeal to several million individual voters, who will not vote as a block.
What we do know is that the last time Labour held a leadership election, union endorsements made almost no difference at all, as can be seen by the lists of formal supporting nominations for the candidates in that three-way contest.
Tony Blair won 52.3% of the affiliated section of the ballot, against 28.4% for John Prescott and 19.3% for Margaret Beckett. The figures for the affiliated section were pretty close to those among individual party members, where Blair won 58.2% with Prescott on 24.4% and Margaret Beckett on 17.4%.
Margaret Beckett had a supporting nomination from the large Transport and General Workers Union.
John Prescott had formal supporting nominations from the following:
Associated Society of Locomotive Engineers & Firemen; Graphical, Paper &
Media Union; National Association of Colliery Overmen, Deputies & Shotfirers;
National Union of Mineworkers; National Union of Rail, Maritime & Transport
While Tony Blair had only the Iron and Steel Trades Confederation - a decade later to become part of the merged union Community, which has nominated David Miliband. (Blair also got rather apt supporting nominations from the Christian Socialist Movement and the Society of Labour Lawyers, as well as Labour students).
So the lesson of 1994 was that nominations had a very weak effect on individual votes in the affiliated section.
However, the candidates are aware that they made rather more difference in 2007, as Patrick Wintour has set out, though the union-by-union figures no longer appear to be publicly available. On the first ballot, Harriet Harman and Hilary Benn did much better among individual members than affiliates, while John Cruddas and Peter Hain did much better among affiliate voters.
Which precedent is more likely to be relevant in 2010?
One might expect endorsements to play a considerably more important role in a deputy leadership contest.
Firstly, there is considerably less national media coverage of a deputy leadership contest, though it is also true that the 2010 race has had less media profile (certainly for any single candidate) than was the case in 1994, where a mid-term contest with Labour so far ahead in the polls made the winner an overwhelming favourite to be the next Prime Minister.
Secondly, there was a much smaller turnout in a deputy race. (From memory, this was as low as 9-10% among trade unions in the affiliated section in 2007, though socialist society turnout - 50% among Fabian Society members - was usually more like that of individual party members). The low union turnout meant that the relative weight of closely engaged union members and activists, perhaps most likely to be engaged with union committee endorsements, would be considerably greater. (My personal guesstimate is that a leadership contest turnout will be at least triple that, and probably higher).
However, perhaps a countervailing difference between 2010 and 1994 is the larger field of leadership candidates, and the more nuanced differences in terms of the political positioning of several of the leading candidates. And if a union endorsement might have some influence on voters' second preferences, that could well matter in 2010 as it did not in the three-cornered first round victory of Blair in 1994.