The UK system is pretty much as described for Canada for all elections, local and national. The local elections (for borough elections) are carried out on a ward by ward basis with each ward having two councillors. The party having most councillors gets to choose the Mayor for the borough but all councillors (of whatever party)have a say in council decisions.

At national level we vote for a Member of Parliament to represent our constituency (which may or may not coincide with the borough depending on population). Each candidate usually represents a party so most people actually vote along party lines. All Members of Parliament have a say in legislation but the administration is carried out by the party with the majority of MPs. What actually happens is that the Queen invites the leader of the majority party to form an administration (this hasn't always happened). This leader isn't obliged to make up the administration from his or her own party and it is possible to have coalition governments where a minority party has enough members to side with another party to block legislation in Parliament.