- Appoint ministers
- Chair cabinet meetings
- Set the agenda
- Appoint senior civil servants
- Award peerages
- Decide when to call a general election
I'd expect most of those powers to continue. However, the governments agenda would in part be set by the demands made by the public through the ID system and partly by national and world events. And without a House of Lords, peerages will not have the same relevance.
With ID the PM would need to keep a watchful eye on public opinion while making decisions because a majority could, in theory, countermand his/her decisions and ultimately call for a general election or the sacking of ministers and senior civil servants. These sanctions would likely only be employed in extremis, but their existence would be an influence on No.10's operations, without voters continuously meddling in the day to day operation of government.