My Boyfriend Cohabits With His Ex

Been seeing my bf for about 6 months now,so getting fairly serious. All going well until recently,when its really started to bug me that hes still living with his ex in the house they took out a mortgage on...
They were together about 5 yrs and broke up about a year ago. Im almost certain theres nothing going on between them anymore,she has another bf now too,but its difficult seeing him text her,know they spend time together etc. Also recently im starting to wonder if shes trying to get back with him,seeing him happy with me maybe. Eg there was an incident recently that ended up with her in a+e,but who did she call at 4am to go to her aid,my bf not her bf. Mine left my house to go help as it sounded serious (obv I understand he must still care about her),but I cant help thinking it was slightly pre meditated to see if he'd leave me to go to her.
Am I right in thinking this situation cant go on much longer? I completely understand the awkward mortgage situation but its now affecting our relationship. How do I go about trying to get him to move out sell up whatever without sounding pushy and controlling??!
Any advice much appreciated :-)