Delta Chat uses OpenPGP for encryption, which does not support forward secrecy.
Apart from backwards compatibility, the greatest obstacle to adding forward secrecy is multidevice support. Double Ratchet protocols (Signal protocol, XMPP OMEMO, Olm, Megolm) do not advance the ratchet on devices which are not participating in the chat, which could be a problem if one of the devices is offline for a long time. Since different devices use different keys, the protocol also depends on the server publishing cross-signatures for offline device keys (for example, via https://xmpp.org/extensions/xep-0060.html in case of OMEMO), which most email servers can’t do without extensions such as Web Key Service.
Another way to implement forward secrecy is to use Off-the-Record Messaging protocol. Unlike OpenPGP and Double Ratchet, it does not support multi-device, but it is designed to be used over existing chat protocols, such as XMPP and IRC. On the protocol level an OTR conversation can be created within existing OpenPGP chat between any two devices.
OTR chat is a separate chat type. User creates an invitation to OTR chat on one of the devices and sends it in 1:1 chat. Once it is accepted on the other side, a separate chat is created which is only available on two participating devices. OTR key could be additionally verified, but it is not strictly necessary as long as OpenPGP key was not compromised and no active attack was taking place at the time when OTR conversation started.
User experience of such chats would be similar to Telegram Secret Chats, which is already familiar to some users.