What should I do if I need import a new private keys in my new wallet but my blockchain is pruned?
Private key is private key and you can import it to a Bitcoin full node or Bitcoin prune node wallet or with even SPV wallet too.
Full node, Prune node or SPV, it is only about how small or less, full or part of Bitcoin blockchain stored on your node and device to verify past Bitcoin blocks and UTXOs. If you don't have a Bitcoin full node, you will have to connect your wallet to servers and nodes that are not belong to you and rely on their operations to get blockchain data.
https://developer.bitcoin.org/reference/rpc/importprivkey.html