Wot's changed? Did it used to motor through, or has it always been reluctant and (in old age) just got more so?
A medium sized drill to me is 3/8 or 1/2 inch capacity, and most "budget" drills will just about do this though they tend to be a bit high geared. Slowing them down by gearing or belt sizes is always going to help, but don't get side-tracked into variable speed if the machine is already on the limit of the motor rating. If you do gear it down will it still be a happy bunny drilling very small holes?
Hopefully the drill has a nameplate on it giving speeds and capacities – if it is rated to drill a 10 or 12 mm hole but has decided not to then I'd look at the drive belts first, as John Haine suggests above, if that looks OK then I fear the motor is poorly. Various things can go wrong with a motor as anno domini's accumulate, but most will give obvious signs of distress by way of hot bits which should run cool, horrible buzzy noises from the motor etc. Single phase or three phase, if it is single is it capacitor run (two capacitors) and if so there's a suspicion that the run capacitor or winding is faulty etc etc. Neil has done a very lucid sketch on another thread in the last day or so explaining the intricacies of a single phase motor with a draggy centrifugal switch.
Check the drive belt isn't stiff with old age (don't I sympathise!) or even beginning to de-laminate. Also bear in mind a lot of these drills used undersized drive pulleys to keep the size of the machine compact. The drive belts don't last for ever going round these little pulleys, especially pulling a large drill at or above the max torque capability of the machine. Experience suggests many of them run a bit fast for the max size of drill specified. A new good quality drive belt can absolutely transform the capabilities of a drilling machine. Are the pulleys alloy, if so have they got a wear groove in the wall of the vee which has altered the drive vee angle? Is the bottom of the drive vee polished, 'cos if so the belt is bearing on the bottom of the vee not on the sides at all.
The other thing that can affect the way it drills is how greedy the drill is, by which I mean how it has been sharpened. A drill with a lot of clearance will grab and be difficult to keep turning, whereas one which is a bit less eager will behave in a more civilized fashion and need less torque to achieve steady progress. Comments earlier about drilling a pilot hole – I use a "rule of two or three" pyramid – so drill a 4 mm pilot for a 12 mm drill, anything over 5 mm needs a pilot hole first. Too big a pilot hole can lead to its own problems, as the drill tends to grab unless it's having to work for its cutting. You can stone a bit of the rake angle off the lip of the drill to calm it down, but you really shouldn't need to unless you're cutting something grabby like green brass or cast iron.
If you can give us more details of the model or construction of the machine we can probably muse further, for example is it pulley and drive belt drive, vee belt or poly-vee, is a variable speed control fitted and if so what speed range do you use, is the motor an induction motor or a brush type. The OP says it is a middle range Axminster Tools machine so we probably guess it's a 1400 rpm induction motor rated at about 1/2 – 2/3 HP, pulley drive with plain vee belts, no variable speed. But confirmation would help. also what speeds are indicated for the different pulley combinations