I think he�ll have to use a bit of the M25 to get across the river. Otherwise it�s up to the Woolwich Ferry (definitely a no-no) or Blackwall Tunnel (almost as bad).
There is a way from J5 of the M25 on to the A25 and this can take you to the A22 (just north of Godstone) where you can head south to join the A264 just north of East Grinstead. The A264 will take you west to the M23.
However, this stretch of the A25 is notoriously bad. It goes through a few (very picturesque but very narrow) villages. I recommend therefore that he stays on the M25 to J7 (M23) and takes the M23 south. The stretch of M25 from J5 to J7 can be a bit congested at times, but nine times out of ten it will prove better than the A25.
Once on the M23 stay on to the end (J11) then take the A264 to its junction with the A29 at Five Oaks. This stretch is dual carriageway for about two thirds of the 10 miles or so.
Stay on the A29 southwards via Billingshurst (new by-pass recently completed) and through Pulborough (not usually too bad). This is single carriageway, but usually moves OK.
Over the Downs to join the A27 west of Arundel. Westwards on the A27, it becomes the M27 near Portsmouth. The M27 all the way to the end (J1 near Cadnam). Then it�s the A31 across the New Forest to Ringwood (dual carriageway all the way) and finally the A338 to Bournemouth.
Whilst using a bit of the M25, this route cuts out the worst bit beyond J7 and avoids the M3 entirely. I use it quite a bit to get down to Dorset. You will always get some traffic on a journey of this length, but generally I�ve found it OK.