" The Policemans Helmet is going to be a bit of a Bother"
He he ...never a truer word Harvey

Are you going to do a fabric Helmet in the same material as his uniform ? Or carve in wood?
If you are doing material , put a tape measure around the wooden head where you want the hat brim to be.
Then divide that measurement into 4 or five equal parts . ( think a helmet of 5 panels would turn out less 'boxy' or 'square' than one made of 4 parts ..but I am only guessing , as I always carve mine.)
Then on some cardboard make a template.
Start off with an equal sided triangle ( but then flesh it out to a gentle arch shape ) ..The base of the triangle is the measurement you got from the head . Find the middle of this line and draw a centre line up from it at right angles ( 90 degrees )
Make the height of this line the height you want your helmet to be plus just under a quarter as much again ( 25 percent ) .....this is because with the curve of the arch , the overall height of the bits when sewn together will shrink down the hat height a bit. Leave a good 25 cm extra at the bottom of each panel to turn under to make a good brim- adjusting the size of this fold will also let you adjust the size of hat to suit too.
Turn these two measurements into first a triangle , and then an 'arch' leaving some of the hat brim ends straight not sloped-- or you will end up with more of a witches hat than a police helmet.
Cut out your templates onto fabric ...add extra for hems all round , and sew them all together and turn inside out. stuff with wadding or foam (personally I would get a 50p bath sponge and snip it with scissors roughly to shape and than stretch some wadding over it to soften out the shape , and put the fabric over that ).
A picture being worth a thousand words , I will try to upload something. Not done this before so hold tight .....

You May not get the shape you want the first time ...in which case adjust the templates and try again until you do ...these things are seldom an exact science.
If you have plenty of your blue material then dive in right away once you have constructed your template.
But if you only have just enough blue cloth to cut out the uniform and the helmet once , then you will have to experiment on some scrap fabric until you get the hat shape working the way you like.
...and yes , I know how frustrating and boring that is when you are keen to see a quick result...so I hope you have lots of blue cloth , and so can afford to dive right in and discard any that do not work ( but you might get lucky first time )
The main problem that could happen , would be getting a great looking hat first time , but then finding it is just a bit to tight to get it onto the puppets head.
This would be because your template measuring was a bit too exact , and your seam allowance not generous enough. You could even try making one of the five panels just a smidge wider than the others to make sure you don't get this problem ( that panel could then be the middle front one you put the badge or button on )
After that there is the question of an actual peaked 'brim' , and should you want one a chinstrap.
I would suggest adding these on after the main hat is made , stuffed and glued / nailed in place.
Others might have templates that include a peak already as part of the design ...but this would mean the whole template had to work perfectly first time ( something probably only achievable if you carve policeman quite regularly like Brian Clarke or Miraiker ..and then only if your puppets' heads are roughly the same size each time )
Doing the extra details as 'extra' add on bits to the bulk of the hat would seem the logical path to me , as that way you can fit the hat to your liking to the head first , before decorating it....and after all that might mean having the helmet at a jaunty angle , or leaning back , or forwards ....and as such adding the brim afterwards would be much simpler .
Also if the main hat was pinned on with tin-tacks or upholstery studs , then the glued on brim etc could be used to cover up the metal fixings ( nice to have the brass ones showing on Punches hat as part of the decoration embellishment - but the Policeman needs a simple look perhaps)
Good Luck .
Richard