Is there a way to limit each branch to only have 1 build running at a time?
You can disable the concurrent builds and configure the maximum to 1, under Administration. You can also override it under the plan settings.
Check out https://confluence.atlassian.com/bamboo/configuring-concurrent-builds-289277193.html for more details.
Oops. Please ignore. Just noticed that you were asking about Bitbucket. My answer is for Bamboo!
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