Sure you can. But unless it's a trivial replacement (this isn't) you need to get it certified, which probably costs as much as paying someone to do it for you.
Unless you can convincingly argue that you did it back before Part P came in (old colour wiring may help). Or don't think anyone's going to notice anyway.
If there's Economy 7, be sure that any immersion heater feed is actually live when you need it to be (as well as having an adequate current rating, obviously). Drilling through the wall to intercept a ring main at a socket (via an appropriate fused spur) may not be a bad option, in terms of minimised disruption.
Tiling though, that's voodoo that is.