Guidance on applying for Schengen and UK visitor visa

The two visas are independent. You can apply for both simultaneously. Apply for the Schengen visa at your nearest Netherlands embassy or consulate, and the UK one at your nearest UK embassy or consulate.
Both visas would have to be multiple entry as you are entering and leaving both multiple times.

This sounds like an interesting cruise route, around the North Sea, I haven't heard of this cruise route before.

You can go from Amsterdam to London by air or train. If you go by train, you go though both Schengen Area exit passport control and UK entry passport control before you board the train in Brussels.

Thx Chris for the quick response - but I do have to submit my passport when I apply for the visa. I don't think I can do it simultaneously
Hence I wanted to get feedback on the sequence in which I need to apply

Thanks
Bhaskar

I don't think the order matters. Neither visa requires the other one to be approved first.
Luckily you don't need your passport regularly (Weekly or daily), because that makes applying for a visa difficult.

Dumb question:

Why do you need a visa?

@evan.dunning, the poster said "(we are permanent residents in US)", which implies they are not citizens of the USA. Therefore I assume they are citizens of a country which does require a visa to enter the Schengen Area and the UK. Judging from the name, I would guess somewhere in the Indian sub-continent (no insult intended if I guessed wrong).

Thanks for that info! i read it as a USA citizen, so should have thought through it more.

Thanks!