In power systems with more and more variable renewable sources, gas generation is playing an increasingly prominent role in providing short-term flexibility to meet net-load requirements. The flexibility provided by the gas turbines in turn relies on the flexibility of the gas network. While there are several discussions on the ability of the gas network in providing this operational flexibility, this has not been clearly modeled or quantified. In addition, the gas network may also be responsible for supplying heating technologies, and low-carbon scenarios see a tighter interaction between the electricity, heating and gas sectors, which calls for a holistic multi-energy system assessment. On these premises, this paper presents an original methodology to quantify the flexibility the gas network can provide to the power system, as well as the constraints it may impose on it, with also consideration of different heating scenarios. This is achieved by a novel multi-stage integrated gas and electrical transmission network model, which uses electrical DC OPF and both steady-state and transient gas analyses. A novel metric that makes use of the concept of zonal linepack is also introduced to assess the integrated gas and electrical flexibility, which is then used to impose gas-related inter-network inter-temporal constraints on the electrical OPF. Case studies are performed for the Great Britain transmission system for different renewables and heating scenarios to demonstrate the proposed integrated flexibility assessment methodology, provide insights into the effects of changes to the heating sector on the multi-energy system's combined flexibility requirements and capability, and assess how the electrical network can experience local generation and reserve constraints related to the gas network's lack of flexibility.