BDH,
The usual kind of mechanical broaching press [Marlco for example] has a geared head and a big capstan like operating boss to wind down a spindle with a toothed rack cut into it.
They seem to command premium prices at machine auction sales and you will do well to secure one under £500 once all the VAT, buyers premium and VAT on that is included. Those additional costs to the hammer price account for about 38% of the sum
Home and Workshop supplies have one on sale at £525 plus VAT plus delivery, this is a 2 speed version used more for pushing 3/8 inch broaches or larger.
For the sizes you are taking about I would extend Jason's experience to include a broach of 1/8 inch, I have pushed those through steel gears for Myford wheels successfully. The job is eased if you have included a suitable size of pilot drill to hog out some of the material beforehand where you can
I am planning to modify a damaged and therefore dirt cheap Chinese hydraulic crane to take me into the larger sizes
Regards
Brian