I went today. Joined the queue at 6.20am between Tate Modern and Blackfriars station, having walked the river route from London Bridge station. From joining the queue at that point to leaving the Houses of Parliament took just under 5 hours.
The walk along the Thames is actually very enjoyable: it follows the Thames Path so the view is constantly changing and there's no time to get bored. If I've got my timings right, Blackfriars to the entrance to Victoria Gardens (next to the Houses of Parliament) was 2.5 hours. Then it's about 2 hours in the 'snake' queue in the gardens, then security and entering Westminster Hall.
Some additional tips:
Forbidden items: includes chewing gum. I don't know about hard sweets (eg mints) but any soft or chewy sweets are probably also for the bin. Liquids also includes gels (so hand sanitiser) and also toothpaste, handcream etc. Not sure about lipsticks - I had a lip balm which they didn't take, but perhaps because there was so little left it didn't show up on the scan. I wouldn't risk taking an actual lipstick as they might bin it. Or perfume.
Toilets: there are portaloos but they don't have hand-washing facilities. The National Theatre have made their indoor toilets available - there were small signs up saying that even overnight the toilets and rest area in the Dorfman Theatre are available. Avail yourself of any opportunity to use a proper toilet and wash your hands! Top tip: just after you enter Victoria Gardens, there's a small set of permanent public toilets at the bottom of the steps and to the right. They're tucked away out of sight and I only noticed them because some women were coming from that direction to join the queue. Separate men's and women's. Only 3 cubicles in the ladies toilet but there was hot water and soap (don't know if it's being replenished during the day). It's supposed to cost 20p to enter (card payment) but the barriers were just left open when I was there.
We got our wristbands near the London Eye but it seems that moved further along the South Bank later in the day. You can leave and rejoin the queue before that on a 'trust' basis, ie rejoin your companion or if you're alone, make the acquaintance of the people in front and behind you and then find them again. But that was only for a quick stop at the portaloos or maybe grab a coffee. By the time we got wristbands there wasn't really anywhere to leave the queue for, apart from coffee shops in the road alongside County Hall. After that you're at Westminster Bridge and then the walkway in front of St Thomas' Hospital and on to Lambeth Bridge.
The walk along the river is actually pleasant, but the 2 hours in Victoria Gardens does become gruelling.