First of all, am I correct in assuming that the Cubs currently have the longest active streak of not winning a championship in any major US sport? (And for that matter, are there any teams anywhere else in the world in a major professional league with a longer streak? I realize this question is confused by things like relegation in the EPL…)
If so, then who will be next in line if the Cubs win?
A former Chicago team. I’m not liking this pattern.
Don’t forget that the previous record was also broken by a Chicago team.
It’s essentially impossible to answer this question with regard to soccer leagues from Europe and South America. Even if you restrict the question to those teams which have at some point played in the top flight (either by starting there, or having been promoted there at some point), there are oodles of them who will have never managed to win the top flight championship. Last year’s English champions, Leicester City Football Club (Go Foxes! #FoxesNeverQuit!) were first promoted to the top flight in 1909 (founded in 1884 as Leicester Fosse), so if we take that as the starting point, they went 107 years before winning the championship (132 years from their founding season).
The longest current drought by a former champion is held by Preston North End F.C., which won the first two titles (1888-89 and 1889-90), and have not won since. That’s 126 years of futility.
Well we know who has the second longest drought in baseball, because they are playing the Cubs in the World Series.
In European Cup football, Benfica and Celtic.
In English football, because of promotion and relegation, there is several different ways you can look at it.
However Preston North End who were champions in the first two years of the Football League in 1888-89 and 1889-90 have not won a championship since, meaning they’ve been waiting 126+ years to repeat their feat. Currently they play in the 2nd tier of English football and have not played in the top tier of English football since the start of the 60s, though have come close a few times to being promoted to the top flight in recent years.
Alternatively Stoke City were founding members of the Football League in 1888 and currently play in the top tier of English football and have never been champions (of the top tier).
I figured there had to be at least one Prem team that didn’t have one under their belt. But notice I already mentioned PNE. 
Besides Stoke, there’s Watford, Middlesborough, Crystal Palace, West Ham, Southampton, Hull City, Swansea and Bournemouth currently in the Prem that haven’t won the top flight. So that’s 9 out of 20 that have never won, unless I screwed up the count somewhere.
Middlesborough has played just about the same number of top flight seasons as Stoke, but they don’t go as far back (1902/03 was their first).
Special mention should go to Bolton Wanderers - they also go back to the founding of the Football League in 1888/1889; they’ve never won, and they’ve played the most years in the top flight without winning - 73 (Stoke and Boro are at 61). Bolton are now in League One after getting relegated from the Championship last year - they’ve had severe financial problems in recent years. They’re currently second in League One so at least their form has turned around.
There will be teams in the Premier League that have never won for as long as that league exists. All the top European leagues have much much less parity than the US leagues.
As a Leicester City FC fan of some length (20 years), I can only say to these clubs:
Keep up the hope!!! 
There used to be someone around the boards who was a big Rochdale fan. Now there’s a team with a record of futility: spent from its founding in 1907 to 2009 having been promoted just ONCE out of the bottom division (third division to start, fourth division after 1959). As a result, the fourth division got to be known as the Rochdale Division/League. 
Happy note: Rochdale got promoted in 2010, went right back down, but came up to third division (currently called League One) again in 2014, and has managed to stick there for the nonce. 
All of which goes to show how disconnected with European reality we are with our franchised league system of sports in America.
Looking more at the Premier League - found another team that has a longer time since last title than the Cubs - Sheffield United. They won the old First Division once, in 1897/98. Not nearly as long a wait as the already mentioned Preston North End, but still longer than the Cubs.
West Bromwich Albion’s only title was in 1919/20; that’s another long wait.
Found another worth mentioning - Dumbarton FC, who are sort of the Preston North End of Scottish football. They shared the title in the first year of the Scottish Football League, in 1890/1891; and won the title outright the next year. They haven’t been at the top level of Scottish football since the 1980s, and as recently as 2008/2009 they were in the fourth level of the pyramid. They’re now back in the Scottish Championship.
Their nickname is “The Sons”. As a longtime American sports fan, one of my favorite things about European football are the team nicknames.
Another thing is, how do you define ‘who’? You can get different answers in American sports depending on whether you’re talking about a team as a corporate entity (e.g. Athletics/A’s) or the fan base in a given city (e.g. Washington, D.C.).
In MLB, the Nats haven’t won a championship since their origin as the Montreal Expos in 1969, but D.C. area baseball fans have no connection with that. Last time a D.C. area MLB team won the World Series was in 1924.
That is a drought. These other examples, of teams which have never won, not. You can’t call a “drought” in a place where rain has never fallen.
From 1903 to 1960 there were sixteen major-league baseball franchises. Their latest World Series wins are:
AMERICAN LEAGUE
Boston: 2013
Cleveland: 1948
Chicago: 2005
Detroit: 1984
New York: 2009
Philadelphia-Kansas City-Oakland: 1989
St. Louis-Baltimore: 1983
Washington-Minnesota: 1991
NATIONAL LEAGUE
Boston-Milwaukee-Atlanta: 1995
Brooklyn-Los Angeles: 1988
Chicago: 1908
Cincinnati: 1990
New York-San Francisco: 2014
Philadelphia: 2008
Pittsburgh: 1979
St. Louis: 2011
To quote Wikipedia
St Johnstone ended a 130-year wait to win a major national trophy (Scottish league championship, Scottish Cup or Scottish League Cup) when they won the 2013–14 Scottish Cup. 2014-130=1884!!!
Also the Vancouver Canucks have gone 46 years without winning a Stanley Cup, the longest drought by any franchise who has never won it.
Which is the year before the current owner and lifelong Washingtonian of the Nats was born. He’s reportedly desperate to win a championship in his life time.
Yeah, those aren’t droughts, those are deserts. 