Is it possible to buy outdoor clothing in Northern India/Nepal?

I haven't been in northern India yet. But my experience of shopping in Nepal has been good so far. Thamel in Kathmandu is a good place to start shopping for trekking gears. You can find cheap copies as well as authentic expensive gear. If you are looking for authentic stuffs, tell the storekeeper upfront. They will take you straight to the authentic items if they have any. There are a few brand stores just a few steps out of Thamel, you can try them out as well.

The stores in Thamel are good for backpacks in general but not so good for foot wear. If you are opting for a full fledged expedition, ask your climbing/trekking guide to help you out. Being locals, they should know the stores better.

Real gore-tex jacket was 4500NR last year. Thermal sleeping bags 4000NR. Hiking boots 2000NR. 45L backpack (Nepali copy) was 1800NR. This is as far as I can remember right now.

How much protective gear you need is subjective. I've seen Nepali porters doing the around Annapurna trek (highest elevation 5400m) wearing sneakers while their clients wore trekking boots. Hope this helps.

If you want to visit mountainous region in Nepal and like to buy some warmer clothes and boots then you'll get it in Kathmandu. Normally but I suggest you to visit Thamel in Kathmandu. There you will get as you prefer for the really region.

When are you leaving for India? I have a trip planned as well and will also be visiting both southern regions and mountainous regions in North India. Perhaps I'll be of some help if I visit it before you start your India trip. I believe that you should buy jacket, hiking boots and similar stuff at an established shop since you will most likely keep those. And you should maybe buy scarfs, mittens, caps, socks and vests at your destination and if you find them quality clothing, keep them, and if not, just leave them for someone else there.

I will update this answer with some prices once I get there.

I wouldn't worry too much about it - for example, Dharamsala is chock full of shops selling cheap outdoor clothing. However, while perfectly functional, they are mostly off-label or pirated Chinese copies, not necessarily the real thing, so I'm not sure if the "quality" will suffice for you. You're unlikely to find authentic branded clothing outside the main cities like Delhi and Kathmandu.

As I write this I'm wearing a fleece bought second hand in Kathmandu.

Plenty of tourists sell warm clothes as they leave or give them to porters who sell them. I would wait for Kathmandu rather than New Delhi.

When I was in India (2002) long term storage of luggage was possible at train stations. I think the cost was negligible.

Considering this, from a cost perspective it might be a good idea to bring your current gear with you and stick it in storage.