In a previous life, I was a removals man. I started out as a Saturday boy and eventually became the owner of a medium sized removals firm. These are a few thoughts.
First, avoid a big firm like Pickfords or Bishops Move; they are basically unaccountable if anything goes wrong. OK, you theoretically have insurance, but try claiming on it if you find something's broken six months after you've moved house. Everyone here only has experience of it going right. Instead, try to get a recommendation for a little guy who has a well established local business. They depend on their reputation.
The problem with packers is you might not find things have been broken (or stolen) for months afterwards. And lots of them still use casual labour. I have hired people randomly from the pub in the past for a day of work. That isn't unusual. Would you really want a stranger going through all your stuff to pack it into boxes?
If you can find a reliable, established little guy then maybe packing is an option. But if it were me, I would dump as much as you can before you move, then just take a long time doing it yourself. It's ultimately the only way to make sure the job is done properly. Ask on Freecycle for proper removals boxes (everyone has them, but they're very expensive unless you buy them by the pallet), and use lots of paper. A couple of hours a day, and it will all be done in a couple of weeks.
I have no idea what a competitive price is for this sort of thing anymore, but to do your move in a day you'd typically use a 7.5 ton wagon and three men.
Hope this helps.