Bat: Cairns
Bowl: Streak
Rest I don't know, both were brilliant though.
You get into Hadlee territory, was Streak truly an 'all-rounder' ? Little doubt Cairns was, not just because of his 33 average in Tests, but because he made five hundreds compared to Streak's solitary hundred. Streak also made only half as many fifties (11) in three more Tests than Cairns
100s & 50s vs 5wi & 10wm
Cairns (62 Tests) - 100 x5, 50 x22. 5wi x13, 10wm x1
Streak (65 Tests) - 100 x1, 50 x11. 5wi x7, 10wm x0
In 215 ODIs Cairns made four hundreds and 26 fifties, Streak played 189 ODIs and made 13 fifties but no hundreds. Not too surprisingly Streak took twice as many 4wi hauls as Cairns.
For me Cairns was an all-rounder, Streak was a bowling all-rounder ie a frontline bowler who was very good with the bat. Hadlee too was a bowling all-rounder rather than a proper one, just two hundreds in 86 Tests compared to NINE 10 wicket matches. If all-round is contributing equally with bat and ball, then neither Streak nor Hadlee contributed anything like as much or as often with the bat as with the ball. They'd both be top of their respective countries all-time bowlers lists, I expect Streak might make the top ten batting for Zimbabwe because of their lack of quality and because they were only a Test nation for a decade or so. Would Hadlee make a top ten kiwi all-rounder list for batting?
I can find only four Zimbabwean Test players with more runs than Streak - Andy and Grant Flower, Alistair Campbell, and Guy Whittall. I'm sure Murray Goodwin, Tatenda Taibu, Neil Johnson, Stuart Carlisle, Andy Blignaut and Sean Ervine were among others not only with a better batting average, but who were better batsmen than Streak. So even as an all-rounder he wasn't the best Zimbabwean all-rounder with the bat, not that all those listed are/were all-rounders but Ervine, Johnson, Whittall and Blignaut certainly were.
As for kiwi all-rounders better with the bat than Hadlee :-
Oram : 1780 runs @ 36.33 (100 x5)
Cairns : 3320 runs @ 33.54 (100 x5)
Reid : 3428 runs @ 33.28 (100 x6)
Congdon : 3448 runs @ 32.22 (100 x7)
Thomson : 958 runs @ 30.90 (100 x1)
Vettori : 3779 runs @ 30.72 (100 x5)
Hadlee : 3124 runs @ 27.17 (100 x2)
Thomson only played 19 Tests, had he played 86 Tests then it would be fair to assume he might have made 4-5 hundreds. Hadlee is the only one to score 3000+ runs and not make 5+ hundreds, Oram managed it without even reaching 2000 runs.
Hadlee clearly wasn't in the same class of batsman as all-rounders, it annoys me when people call him that. I think it stems from the era when Imran Khan, Ian Botham and Kapil Dev were greats and great all-rounders and, because Hadlee was no mug with a bat, he was thrown into a four-way comparison even though he was clearly a bowler and not a batsman. Malcolm Marshall was another who wasn't really an all-rounder, he could bat but never made a hundred.
All-rounder is a term branded about willy nilly by fans, referring to the fact a bowler can bat or a batsman can bowl. But if you are comparing "all-rounders" you do need to check they are comparable and it is reasonable to call them all-rounders. If anyone ever quotes a rating system for all-rounders, throw them a 'curve ball' and ask them to insert a batsman (Bradman) or a bowler (McGrath). They should be filtered out by minimum requirements, at least one 5wi and one hundred would be a start, not to mention 100+ wickets and 1000+ runs if they are doing "greatest evers". The rare oddity like Gillespie might beat most filters, but Bradman and Walsh shouldn't. The flaw in many rating systems can be exposed by Bradman because his batting average was so high, and his bowling average was 36 (two wickets). If that isn't filtered out it is a poor system. I've thrown in McGrath as a bowler averaging under 22, 500+ Test wickets and no fifties to his name.