Are my sash windows unusual, I can't work out how to reglaze it?

That's how they are.
Knock out all the broken glass and then use a thin chisel to clean out the rebate, there wont be any sprigs in there.

 

I take them out of the box to reglaze, as its difficult to work them in the box. but that can also lead to having to replace some beadings, cords and repair work to sash woodwork or box. also you may as well renew the cords and make the box and sashes effectable.

the putty could be like stone and not easy to get out. so i put it in a vice upside down and clear out the rebate. occasionally using a heat gun helps.

having done that its simpler to take to a glazier and get them to fit and putty, or even do the clearing as well. make sure they use the same thickness glass or you will have to change the weights, another cost.

there isn 't a one stop shop to do all this, so either getting about or buying tools and materials and such is likely.

 

What is your location?

 

This takes me back a bit, its a while since I have done any repairs to sliding sash frames.
First of all Mr B, I would assess the state of the frames for rot etc, do they slide up and down ok? Its an ideal time to renovate and re-cord if necessary.
The windows that I repaired over the years the glass was bedded in to the top rail of the lower sash with good soft putty by pushing the pane into a groove, not a rebate. This groove was around 6mm deep, 1/4 inch in old money; so when ordering new glass an allowance has to be made to allow the glass to be placed and pushed up into the groove, but also needs to sit in the rebate on the bottom rail of the sash. Be careful to order the same thickness of glass otherwise it will not fit into the groove, and as Chippie244 states the weight of the sash needs to be counterbalanced by the sash weights in the pockets.
Use a Glaziers hacking knife, cheap enough from a glass merchants to remove most of the old putty followed by a really sharp chisel to remove what's left, then seal the rebates with paint or similar to help stop the oils in the new putty from soaking into the timber which can cause the putty to crack.
If you intend to repair the window yourself including the putty pointing I would advise you to buy a putty knife to make the job a little easier, and buy a good quality putty that hasn't been sat on a shelf for years. It used to be my responsibility to turn over the tubs of putty regular and knead the putty ready for use.
I used push points to hold the glass in place instead of nails as nails rust over time and push points are slid into place with a chisel across the face of the glass into the rebates, this reduces the chance of the glass breaking using nails.