Storing luggage

Pack less, do laundry more often. Then you won't need that extra suitcase.

There is a left luggage at St Pancras station: https://www.left-baggage.co.uk/en
Plus this one: https://www.citystasher.com/your-guide-to-left-luggage-in-london
But it ain't going to be cheap.

Heathrow to Amsterdam, Lucerne, Edinburgh and then return to London

That is a bit of zig-zagging. What would be more logical is London - (train)Edinburgh - (fly)Amsterdam - (fly Zürich airport)Luzern - (fly)London, or vice-versa. Or even better (apart from your bag problem), fly home direct from Zürich.

There are left luggage facilities at each terminal at Heathrow, so you could get rid of the large bag at the airport when you fly to Amsterdam, and pick it up on the way back. I expect it will be costly.

Try citystasher.com. It lists businesses and other places that will store luggage for you for a fee. It's fully insured and the prices are generally better than a regular left-luggage place.

Yes, as suggested you could leave this luggage at the left luggage facility. But for three weeks I would carry this luggage outside of London instead of using the left luggage option. You're only going to three other places. What if you have another 3 weeks added to this trip?

Some hotels will keep luggage, I know that one hotel kept my daughters and her friends for two weeks while they travel around Europe, then they came back to England picked up luggage and went to Scotland for a month.