Thanks to everyone who replied. You've given me some more ideas of how to tackle it. Just for info this is an old bike my son has and he took the photo. I've always said that anyone who has a bike should know how to fix it. He has finally begun to take that to heart but clearly didn't clean it up before he started!
It may take us a lot of effort to fix but it will be a learning experience. And there's no point him having a fancy bike as it will just get stolen.

I don't know why you can't just use a hammer and a punch, drift or cold chisel to get it out. Especially if you aren't going to use the BB again And it doesn't even need to be a big hammer. Just put the punch at 90° the edge of one of those splines so you are hammering in the direction it needs to turn.

Just make sure you are going the correct direction. Left side has right threading, right side has left threading..... or is it the other way? You should check! Some oddball BB's are different.

If you are rebuilding the bike from the frame up and buying new components, you really want it to be a worthy frame. Otherwise you are putting lipstick on a sow. And could have bought another bike for less that needs nothing done to it.

But for learning experience, it might be reasonable regardless of the cost.

My "Modern Bike Repair, 101" Class was a Trek 720 MTB i bought for $45, then discovered that it needed just about EVERYTHING rebuilt or replaced... i spent more on tools than parts... the experience and education is Priceless.

i still have all of those tools.. except the chromed 13mm Park cone wrench i snapped getting a mongoose BMX front axle loose. Park Tools sent me a new one after i emailed them a pic of it,and assured them that i had broken it bare handed, without a helper bar.. the pic included a view of my right forearm.

Hi everyone. Just thought I'd report back for your interest.
Ridgeback, the bike company, tried to be helpful but the company had changed hands so many times that no one had experience of the strange bottom bracket except to say that it was "probably Chinese" Local bike shop also had not seen anything like it.
Once I got to look at the bike myself it turned out that the drive side had a regular set of flats on the external diameter of the BB and could be removed easily with an impact wrench. Inside was an old style set of cups with cones on the shaft and loose ball bearings. So far so good...
However, the non-drive side (as per the picture) was not shifting. We tried hitting it with punch and 4 pound hammer, heating it up and letter it cool, gripping the small amount of thread protruding in a vice and turning the bike, all to no effect.
As advised on this forum we then tried the 'bolt' method. Initially we used an M10 set screw that I had lying around with a stack of washers. Rather than the BB coming loose we rounded a few nuts and extruded a few washers into nice cone shapes! But it did seem like this was the only chance to loosen it.
Finally we realised that an M16 hex head should fit with its corners poking into the strange 6 spline indent on the bracket. So I bought one M16 setscrew from a hardware shop and pulled it tight against the bracket using an old box spanner instead of the washer stack. (see photos in album "unusual bottom bracket"). This tightened up well but wouldn't turn the BB even with a yard long scaffolding pole on the spanner on the drive side nut. But as the bolt head was now nicely jammed into the BB indents on the non-drive side we decided to put the spanner on the non-drive side and hit it. Finally it moved! It never came loose until almost the whole way out but strangely there wasn't much sign of corrosion on the threads.
We cleaned up the threads and inserted a standard BB with plenty of grease. Both sides screwed in pretty much all the way by hand.
So if anyone finds one like that again, get yourself an M16 bolt!