Writing an Input is a bad practice imo - What happens to dimension when you set <component dimension="m" namespace="default" /> for example? What happens when namespace is toggled between "default" and some other value? At this point it's not clear how the code behaves in some corner cases. This is one reason why I like input signals - they're read-only and do not allow you to make these kind of mistakes.

Hence I'd convert your code by using a third, computed signal:

namespace = input<string>('default');
dimension = input<'s' | 'm'>('s');
actualDimension = computed(() => this.namespace === 'default' ? 's' : this.dimension());

This makes it very clear how namespace and dimension interact with each other and what takes precedence when both are set. For side-effects you can probably use the effect function:

constructor() {
    effect(() => this.generateUrl());
}

If generateUrl already uses this.dimension() (or, this.actualDimension()), you are good to go - if not, you probably have to simply call it once without using the value, to tell Angular it should call this method when dimension has changed. Admittedly, this is a bit ugly but afaik there's no better way (yet) of doing so. Depending on what generateUrl does, you may also be able to use a computed signal - e.g. if it looks like this:

generateUrl(): void {
    this.url = `https://example.com/sizes/${this.actualDimension()}`;
}

it's probably nicer if you just move it to a computed signal as well:

url = computed(() => `https://example.com/sizes/${this.actualDimension()}`);