If you have points earned from the last league you can still use them (both for old and new rewards). Your new points gained in this league can also be used to obtain the last league's rewards and of course the new ones.
This thread was added on August 14, 2020, with posts from JagexAcorn.
Original Post
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If you have points earned from the last league you can still use them (both for old and new rewards). Your new points gained in this league can also be used to obtain the last league's rewards and of course the new ones.