Regardless of destinations of traffic, the whole point about the bridge rebuild/heightening is that standard double-deckers can get from the city centre out west (note the current bus station, and the new planned bus interchange are both on the east side of the rail line) - those services include local buses to the Park and Ride(!), services to Cumnor and Botley, and then the ones that go further afield like the 66 to Faringdon/Swindon, and the 4 to Wootton/Abingdon (the long way round). For the majority of those services, if you close the road, and make them go north or south first, in rush hour it could easily make a half hour trip at least twice as long.
If you're talking about allowing only buses/taxis etc, then good luck with that. Having a small (sensible!) road closure in Jericho to create an LTN and stop rat-running through there to avoid Woodstock Rd has caused enough arguments in the last year - and that's without a large number of people in work in Oxford!