The first thing to note is that the IWM will be closed throughout the first half of 2013:
http://www.iwm.org.uk/visits/iwm-london
Losgigs seems to have provided the wrong link. This appears to be the correct one:
http://www.iwm.org.uk/visits/iwm-london/access
Reading station offers step-free access to all platforms, via the lifts. Pre-booked assistance can be arranged by phoning 0800 197 1329.
http://www.firstgreatwestern.co.uk/Your-journey/Assisted-travel/~/media/PDF/About%20us/DPPP%20Customer%20DL%20aw.pdf
Trains to London Paddington take about half an hour. Seat reservations are available on the majority of services (via the above phone number or Customer Services on 08457 000 125).
If you've pre-booked assistance, staff at Paddington can meet you on arrival to assist you.
One possible option is then to take bus 159 (towards Streatham) from stop H outside the station. The journey to Kennington Road (very close to the IWM) takes around three quarters of an hour.
Use the journey planner at the top right of this page to seek other routes (paying particular attention to the 'Other options' facility, which lets you select your mobility priorities):
www.tfl.gov.uk
(Note that although 'Imperial War Museum' isn't offered as a pre-selected option, it's still recognised when you enter it).
Chris