Thank you. If he does not pay rent then he is an excluded occupier and can be evicted without a court order. Ideally, you would also not have given any kind of exclusive occupation - e.g. you continue to come and go as you please and out of the property to carry out for example your administrative duties.
To remove him from the property, you need only give him notice which as I understand that you have already done, and then you can simply attend with a locksmith to change the locks whilst he is out if you wish. You would need to act in concert with your co-executor as you must act jointly in such matters. You cannot remove him with force.
in terms of a potential claim he may have against the estate, there is the possibility of a claim under the Inheritance Act. This provides that a person
"who immediately before the death of the deceased was being maintained, either wholly or partly, by the deceased" may make a claim for reasonable financial provision. However, what is key here is the words "immediately before". Accordingly, he would need to be able to show that he was being maintained which could include living rent free in your mother's home or been paid money by your mother in the immediate period before her passing. Unless she was paying him money at this time, the fact that he had moved out with on the face of it appeared to make such a claim difficult. In any event, any such claim would need to be made within six months of grant of probate.