We did this journey in reverse a few years ago via the Grand Union/Stratford/River Avon route, It took us 16 days, we were not pushing it but we were/are experienced boaters so as inexperienced boater I would allow at least this amount of time. if not longer.
This trip will soon turn you into experienced boaters! Steering a boat and using locks is not rocket science. Don't be afraid to ask other boaters advice, they are generally a good bunch but expect sneers when saying that you are taking the boat to London.
The Avon and Worcester and Birmingham routes are about the same in miles but the Avon route has 189 locks as opposed to the 213 of the W & B Route, to Little Venice, add another day and 12 Locks to get to the River Lee. The disadvantage of the Avon route is having to buy an additional licence (£50) and that the first couple of days will be all River Locks which take a bit more nous. This route might not be the best when you are still learning about your boat. There is also a stoppage on the Sothern Stratford until 15th March but as the stoppage at Gloucester Lock shows no sign of coming off this may not be a problem!
As has been mentioned your daughter should join the London Boaters Facebook page ASAP as it is an excellent source of information and a good self help group.
There is a tremendous boating community in London and is a very cheap way of living here but it does come with 'inconveniences' such as having to find a new mooring every couple of weeks and having to empty toilets, buy fuel etc. Getting maintenance/repairs done outside of marinas can also be problematical. Trying to continuously cruise just on the River Lee will be problematical. Not only will you be expected to move every two weeks but you will also be expected to cover a rage (20miles?) over the year so shuffling from one mooring spot to another nearby one will soon get you into trouble.