Some of the objections are that it will be a costly project which will end up with expensive tickets
Most are simply from people in the wealthy north-of-London commuter area and expensive parts of the country who won't see any benefits personally (they can get to London easily) but it will pass near them.
For those in the West Midlands, North West, Scotland and later on the North East, it will make a huge difference to travel to London. It won't shorten a single journey to London by 30 minutes, it will more than halve my travel time to London from over 2 hours to under 1. It will shorten 5 hours to 4, 4 to 3, 3 to under 2, 2 to less than 1, and will make Birmingham a quick jaunt from London. Joining our two largest cities with a 30 minute service can only be a good thing. Putting Manchester an hour away isn't much worse - business in the UK will love this.
It will also have other effects, it's not just about cutting down the direct London-Birmingham/Liverpool/Manchester etc journeys, it will free up a lot of capacity for local and shorter express services. It should also help punctuality - right now a single delay on the WCML can have knock-on effects for hours. HS2 should give more head-room for catching back up to timetables, not to mention capacity for diversions (meaning less rail replacement buses)
Perhaps more importantly it will free up a lot of desperately needed freight capacity - if we offload all the long distance, nearly all the middle distance and some of the local/commuter trains from the West Coast Main Line, we will be able to put a lot more freight on rails instead of the roads. This will be cheaper, greener and will un-clog various motorways and A-roads, not to mention taking lorries off the roads into city centres.
Parts of the West Coast Main Line are running at absolute capacity, most of the East Coast Main Line and Midland Main Line aren't far behind. Very simply, we NEED more north-south rail capacity. We may as well do that with a high speed, modern line.