House training is agony, but if you want to get it cracked asap, you just have to resign yourself to:
a. NEVER letting the puppy out of your sight, unless it's in a crate, for several weeks
b. taking it outside after meals, when it wakes up, after playtime, and then on the hour (prob half hour for smaller pups) so ideally it never gets a chance to wee/poo indoors. Sniffing is a good sign, but their bladders are tiny, and you don't always catch them in time, and then you're left carrying a weeing puppy across the floor to the garden like some kind of furry water pistol.
c. giving wild and enthusiastic praise when it does wee outdoors, so it gets a clear sign that it's the right place. Try to yell a trigger word while it's in action, so eventually you can take them out, mutter, 'wee, please' or similar, and know they'll perform promptly. I kept a treat jar outside, so I could reward on the spot - our dog's very food-orientated.
I was advised never to use paper indoors, because it just confuses the puppy. It sounds rather anal (hee), but I made a note for a few days of when my puppy needed to go outside, so I knew roughly how long I had between trips, then tried to anticipate so neither of us got caught out.
Obv this is all much easier said than done. I went through a whole gallon of Simple Solution and hundreds of kitchen rolls, and that was just from me weeping tears of despair as my carpet was soaked again. But my puppy was watertight by five months, and he's a notoriously slow-to-housebreak, happy-in-own-filth breed, so it won't take your Westie half as long if you're vigilant.