See page 19 of the rule book under the heading "Playing Mice and Mystics as a Campaign":
1. Mice keep their story achievements and abilities earned from previous chapters.
2. Mice can keep only 1 search card that they had at the end of the previous chapters. This 1 card is in addition to their starting equipment cards. Mice will always begin a new chapter with their starting
3. Mice do not keep any party items, individual achievements, or cheese from previous chapters. (emphasis added)
So by rule, Tilda's cheese is lost. You're free to house-rule of course. Just be aware that this will almost certainly affect the intended game balance.