Flats are different to houses, a house owner has control (and responsibility for) everything up to their boundary but that isn't the case in flats where services, roof, common areas etc are shared. Then it's the responsibility of the managing agent. They won't work at weekends and need to get more than one quote so it's likely to take longer to organise. If it's something expensive there are additional hoops to jump through with the owners.
Sewage backing up is a health hazard though and needs to treated as an emergency, I'm glad the council are acting swiftly. Probably payment will be through the service charge the landlord pays. In any event he is responsible for plumbing and drainage so the tenant shouldn't pay for it. The landlord is paying the letting agent to manage his property and IMO the letting agent hasn't done his job properly. If it was his own house I'm sure it would be fixed by now.
