That would require dual voltage stock (which would perhaps be available following Basingstoke-Soton AC conversion), and lengthen journey times by at least ~40 minutes. It'd be easier to shuttle passengers to Reading for connections to London (with most choosing Paddington, obviously). WoE services could do this already, which I've never seen, so I guess there's good (well, maybe not so good) reasons for this.
I suggested post Reading to Basingstoke electrification as I was thinking that there would be duel voltage stock by then (or very shortly after).
A increase in journey time of 40 minutes could be shaved down a bit by a running skip stop calling pattern, so possibly up to 30 minutes longer, which makes it not too bad compared with getting on a bus. Yes a lot of people would rather go into Paddington, but having to only make one change at Reading (rather than changing at Basingstoke as well) would be much more prefrable.
WoE services could do it, however I thought that the underpass east of Reading had been out of use. Even if it was able to be used I'd dread to think of how much disruption would be caused by more trains crossing the main lines at Reading (basicly the reason for the remodeling).
Also the class 158's and 159's are fairly heaverly used as it is without them having to do a whole load more work (as they'd have to run full length from Basingstoke) and only have fairly short runs between stations which would be a lot slower than an EMU doing the same trip. Given if there were engineering works there would likely be a lot more EMU's (as they wouldn't need to run at their full length as it was a weekend) that could be spared to run the service (even if not all the fleet that needed converting had been done so yet).