I've always had some problems with this kind of sentence (past + would) in English, so I hope I got the meaning correctly: in the past, Mary said that the child would probably fall asleep soon, but "fall asleep" is in the past itself, i.e. the child has already fallen asleep or in any case is not expected to fall asleep now.
Italiano
Maria disse che il bambino si sarebbe probabilmente addormentato presto.
/ma'ria̯ 'disse ke il bam'bino si sa'rεbbe probabil'mente addormen'tato 'prεsto/
Maria disse che il bambin-o si sar-ebbe probabil-mente addorment-ato presto
Mary say.REMOTE_PAST that DEF.MS child-M REFL REFL-COND.3SG probable-ADV fall_asleep-PART.PAST soon
Mary said that the child would probably fall asleep soon.
È probabile che Maria abbia detto che il bambino si sarebbe addormentato presto.
/ε pro'babile ke ma'ria̯ 'abbja 'detto ke il bam'bino si sa'rεbbe addormen'tato 'prεsto/
è probabile che Maria abbia_detto che il bambin-o si sar-ebbe addorment-ato presto
be.3SG probable that Mary say.SUBJ.PAST.3SG that DEF.MS child-M REFL REFL-COND.3SG fall_asleep-PART.PAST soon
It is probable that Mary said that the child would fall asleep soon.
Notice that the second sentence uses the subjunctive (because we're not sure about Mary saying that), and both use the conditional in their second clause, much like in English.