Two new organisations need to be created to formalise and eventually commercialise the government's role as 'lender of last resort' and 'insurer of last resort' to financial institutions. I'll call these 'Bigbank' and 'Biginsure'
Bigbank's role is to lend other banks so long as the other bank is solvent. Bigbank's lending should be to other creditworthy institutions. It should it insure the loans it makes against the risk of default where possible, by buying credit default protection in the open market. Bigbank's role is to provide 'bridging loans' to other banks who might have temporary liquidity problems due to speculative attack. Bigbank could (say) have around £100bn of capital. The government providing the initial capitalisation, by buying shares, or by guarantee. So long as the risks on the loans that it makes can be adequately insured, Bigbank can lend many times its' capital.
Biginsure's role is to be an 'insurer of last resort'. Biginsure will insure bigbank against it's counterparty going bust, if no other insurers can be found. Big insure could also insure against other major risks where at present. the government provides a guarantee (e.g. terrorism, natural disasters). Biginsure would be well capitalised, in order to cover quite large potential risks. Biginsure could grow (e.g from an initial size of £100bn of government-owned shares to a size of £1tr) by selling shares to the public, so spreading risk broadly.
Both institutions can act commercially, by providing loans at rates that account for the inherent (although not the speculation-induced) riskiness of lending (because now-solvent institutions might become insolvent in the future). The main role of bigbank is to offer liquidity and thus defend solvent institutions against speculative attack.