Cheaper, but far fewer benefits.
Woking to Old Oak Common would be less than 37 minutes (16 minutes Woking to Heathrow and 21 minutes current Heathrow Express to Paddington, although Old Oak Common should cut 4 minutes from that, so about 33 -34 minutes), Woking to Clapham Junction is 19 minutes with then Clapham Junction to Shepherds Bush being 12 minutes, which is already 31 minutes, and so by the time there's been a change of trains at Clapham junction is almost certainly going to be longer.
However, that relies on the already fairly busy services from Woking towards London Waterloo.
There's also the fact that you're putting all your eggs in one basket, so for example this weekend the fastest journey time to Clapham Junction is 48 minutes, leading to a journey time of an hour.
Anyway, the conversation was in the context of Heathrow funding it, in which case the cost is of less concern.
Woking to Clapham Junction isn't 19 minutes during the Monday to Friday morning peak rush hour when trains skip it. After the 07:13 there isn't another until 08:46.
In-between you have to change trains and the times are officially as follows:
46m [departing 07:32]
1h05m [departing 07:46]
56m [departing 08:02]
That is ignoring any that get overtaken and I've told it to avoid Waterloo, as it tries to route you via that route, which whilst faster might not be valid.
The 07:46 is the 1h05m one and that gets you in just 14m before the 08:46.
The 08;02 gets you in just 7m earlier than the 08:46.
So routing people via Clapham Junction isn't much help at that time unless you live in Greater London or near by on the right railway line.
In the evening for the reverse the gap is just 1 hour between 17:02 and 18:02 if going to Woking. Those taking 27m and 23m respectively. There is of course the 17:29, which is non-stop at 41m but it isn't fast. At least it gets in at 18:10, if it's helpful for connections.
So avoiding Clapham Junction would be good, if you are interested in season ticket holders or anyone who doesn't want central London.