Both Batman and Batman Returns are rated 15 by the BBFC. Batman Returns was cut for its rating. Both films were rated 15 for home video and non theatrical viewing (on-demand). At the time Batman was one of the first 12 rated movies in the UK, but they didn't bring in a 12 for home video until a few years later (so it was a 15). Batman Returns was cut for it's 12 on home video, but re-issued as a 15 later on for it's DVD and Blu-ray release.
The difference with the Nolan pictures is that whilst they contain more violence there's very little consequence or injury detail to it, so it gets by with a 12A or 12 on home video. The first two Batman films by Tim Burton are quite dark. After complaints about Returns there was a conscious effort by Warner Bros. to lighten things up a bit as they were aimed at children with their merchandise, but quite unsuitable for younger viewers.
If you have younger children, I'd recommend watching the films with them or watch them first and judge if they're suitable for family viewing. Batman Returns can be quite frightening for younger children.
