In the bitcoin data directory, there is a file called fee_estimates.dat. This file is only written when bitcoind or bitcoin-qt exits so it is not up to date until that point. After that you could copy it and use it on another node to start with up to date estimates.
There is some trickery if the nodes are at different chain heights, but it should mostly do the right thing. It tracks the block height that the estimates were written as of and the new node won't affect the estimates until it's active chain has advanced beyond the latest recorded block in the old file. Also if the chain in the new node is past the last recorded block in the file, then the estimates are decayed under the assumption that the data is no longer as meaningful.
In general if you start from scratch, you should only need to wait for about twice as many blocks as the target you would like an estimate for before you are able to get a somewhat reasonable answer.