Paying for contents insurance in sheltered housing...

My elderly Mother resides in Sheltered Housing for the elderly. As part of her rent and service charge for the past 15 years, Contents insurance has been included. Unfortunately, the Landlord has been told that they cannot provide this valuable service. The premium was just £2.00 per month for about £25,000 of cover for personal possessions etc. My question is: If my Mother has contents insurance, but an upstairs neighbour doesn't. If the neighbour say, leaves a bath filling up and it over flows causing damage to my Mother's possessions. Would my Mother's insurance company seek to recover costs and compensation from the neighbour, personally? or would the landlord be responsible? (Retirement housing). My Mother is naturally very concerned, as it would appear many tenants just wouldn't be bothered with buying insurance, some overlooking because it has always been provided within the monthly rent/service charge, some just wouldn't bother adopting an attitude 'I don't have any money so I can't pay compensation-sue me'. I understand that the Landlord seemingly wants to provide this extremely economical block insurance cover for tenants as part of the service, but has been told they can't charge for this service.