New builds won't be a major issue - like you say heat pumps and electric cookers will be built in from the start. (Do these houses have any form of heating in a power cut?). ....
No more than a house with a normal central heating system does now. If the mains goes off, the gas valve in the boiler and the pump won't work. On a more constructive note though, with the proliferation of EVs, from about 10 years there is likely to be increasing availability of hand-me-down batteries which have been put into use as buffers for peak demand electricity.
... Replacing existing systems on the other hand is not a small job. Substantial subsidies will be needed to drive mass adoption of heat pumps, and cookers are only likely to be replaced when the whole kitchen is. The problem with trying to use price rises to force people to abandon gas is who you are targeting. Early adopters are usually much better off than the average consumer. Certainly you can't start putting prices up until social housing has switched, and without a comprehensive public investment programme the last gas users will still likely be people in poorly insulated homes unable to afford new systems or reliant on uninterested landlords.
The first action is to define a (much ) improved insulation standard for all new dwellings. This could be done almost immediately against a (say) 5 year enforcement date. The planning process would be sensitive to applications that didn't fully comply with all the relevant standards. The additional cost of implementing the thermal performance improvements in new build would be insignificant, and the resulting asset woulld be cheaper to live in.
The first physical priority is a programme to increase the quality of insulation in as many existing homes that need it. Rather than the farcical 'green deal' which cynically didn't want anybody to get 'something for nothing', the administration needs to view domestic energy efficiency as an investment as part of their commitment to decarbonisation of the country.
Starting with the most urgent cases, - poorly insulated social accomodation, where residents are closing every window to reduce draughts and in the process, sealing in harmful fungal spores to the detriment of their health. If a holistic view is taken, preventing that provides a substantial improvement in the health of that part of society with it's attendant hit on national health and welfare costs. If it means additional costs for LAs, then so be it, central government makes it available as ring-fenced capital.
In the case of private rented properties, there should be a carrot and stick approach, i.e some assistance in the costs to bring properites up to defined standards (which should be linked to a licensing system), but a penalty if no action is taken - ultimately removing the right to let sub-standard properties. Such a scheme should protect tenants from any costs that the landord incurs intended to increase the market value of the property.
If all that can be underway over 15-25 years, there would be a reducing trend in the domestic heating energy requirement which would make matching the additional investment in electric distribution to the increase in load more viable. That sort of period is also compatible with the typical lifecycle of kitchen and boiler replacement. The white goods equipment industry would respond with items that are designed to work in the new environment, - they would need to as the existing fossil fuelled kit wouldn't be attractive with higher gas prices.