The most likely candidate to control both canals would probably be Britain. If Britain controlled the Panama Canal (in addition to Suez), it probably wouldn't be too different from OTL, since Victorian and Edwardian Britain both promoted a laissez faire, free trading global economy, so wouldn't interfere too much as long as Britain controls the sealanes.
Even during war time it wouldn't make much difference from OTL (assuming the UK has the same enemies), since enemy nations like Germany would have little influence, trade dependency or power projection in the Panama Canal and would be blockaded by the RN anyway.
If another power controlled both canals, then things would be very different.
The US might be able to control both canals post WWII, i.e. they make an agreement with the UK or Egypt over Suez, so they control both. This would lead to more tension with the USSR during the Cold War and anti-American Arab nationalists swaying even further towards the Communist camp.
France had an influence over both canals, but you would have to butterfly away British naval supremacy to enable them to control them.
Any other country controlling the two canals would require significant PODs going back a long way.