With Bamboo Cloud you were able to set up a pretty convenient "intermediate" solution, with Bamboo Cloud + an agent on your servers. Will it still be possible with Pipelines?
Also I couldn't find the doc for aggregating tests results.
Not so much. One of the things that I think makes Pipelines better suited for cloud is that it's agent-less. But that does mean there's no option to run an agent on-premise to bridge pipeline execution. Indeed, if you are accustomed to Bamboo, you are likely to find Bitbucket Pipelines rather minimalist. For example, there is currently no facility for aggregating test results.