if + should / if + will / if + would

If + should

We can use if with should to refer to events which might happen by chance or by accident:

If you should bump into Carol, can you tell her I’m looking for her? (If by chance you bump into Carol.)

If the government should ever find itself in this situation again, it is to be hoped it would act more quickly.

Conditional clauses with will or would

Will and would can be used in conditional clauses, either with the meaning of ‘being willing to do something’, or to refer to later results:

If Clare will meet us at the airport, it will save us a lot of time. (if Clare is willing to meet us)

If you would all stop shouting, I will try and explain the situation!

If it will make you happy, I’ll stay at home tonight. (If it is true that you will be happy as a result, I’ll stay at home tonight.)

We sometimes stress the will or would, especially if we doubt that the result will be the one mentioned:

If it really would save the planet, I’d stop using my car tomorrow. (If it really is true that the planet would be saved as a result, I would stop using my car, but I doubt it is true.)

Conditionals in speaking

In speaking, we often use if-clauses without main clauses, especially when asking people politely to do things. If is usually followed by will, would, can or could when it is used to be polite:

[Shop assistant to customer]

If you would just sign here, please. (a more polite way of saying Just sign here, please.)

[A is writing something for B and having difficulty]

A: If I could have a better pen …

B: Here, use this one.

A: Thanks.

(from https://dictionary.cambridge.org/grammar/british-grammar/conditionals-if)