Not really, almost all of the things done by government for the daily management of the country are not related to the constitution and are already codified.
You would need to figure out, agree and codify a mechanism for changing power, but you can arrange it so that you have years to do it and codifying that small part is not that huge of a task; USA had long conventions but that was more because of the need to actually create a country; and many other countries post-revolutions have functioned quite reasonably while codifying the constitutions, because it really doesn't affect the daily things, and you can run on interim rules (in UK's case, continuing as before for a limited time) until it's done.
"The Basic Laws of Sweden (Swedish: Sveriges grundlagar) are the four constitutional laws of the Kingdom of Sweden that regulate the Swedish political system, acting in a similar manner to the constitutions of most countries. ... Together, they constitute a basic framework that stands above other laws and regulation, and also define which agreements are themselves above normal Swedish law."[1]
You're right, though, that sounds a lot like the arrangement that Israel has, and the Wikipedia article for the "Basic Laws of Israel"[2] says (without citation):
"Israel is one of 6 countries (along with Canada, New Zealand, San Marino, Saudi Arabia and the United Kingdom) that operates entirely or in part according to an uncodified constitution..."
[0] https://en.wikipedia.org/wiki/Constitution_of_Canada