There is a database for post 1858, but it's not online yet.
I was up the Probate office in Holborn last week. They have a computer there that you can search Wills from 1920 to date, but they give a warning that it is not 100% accurate,they recommend you check the books as well.
Also at WDYTYA the Probate department had a searchable database which allowed searches from 1858 to date, which showed scans of the Probate books, again they said it wasn't error free, but they were saying it should go online soon.
I suspect they are wary of the extra work they will get from this. I'm sure many will want to order their relatives Wills if it was easier. When I was there last week I waited 3 hours for their one hour service. I've waited longer, finally resorting to having it posted to me the next day! With an online ordering service I think they will be swamped.
Oh for the days when they were at Somerset House. Maybe rose tinted glasses but I seem to remember it took less than half an hour for them to produce or copy the Will for you, for only 25p per sheet. When they moved to Holborn I believe it went up to £5.