No sé si l'oncle ho sap, però espero que no.

There are two things going on here:

• Catalan normally uses the definite article with many family nouns when referring to a specific relative: l'oncle, la mare, el pare, etc.
• el becomes l' before a word beginning with a vowel: el oncle → l'oncle.

So l'oncle is just the regular contracted form. In direct address, though, the article is usually dropped: Oncle, vine! = Uncle, come!

Here si means if / whether and introduces an indirect yes-no question:

• No sé si... = I don’t know if / whether...

So No sé si l'oncle ho sap means I don’t know if the uncle knows it.

This is different from English in the sense that Catalan uses si very naturally after verbs like saber, preguntar, veure, etc. to report a yes-no question.

Ho is a weak pronoun, and here it means something like it or that.

In particular, ho often refers to:

• an idea
• a fact
• a situation
• something previously mentioned but not treated as a masculine or feminine noun

So:

• ho sap = knows it / knows that

If the thing referred to were a specific masculine or feminine noun, Catalan would often use a different pronoun, such as el or la.

In Catalan, unstressed object pronouns usually go before a conjugated verb:

• ho sap
• la veu
• els coneix

They usually go after the verb only with:

• infinitives: saber-ho
• gerunds: sabent-ho
• affirmative imperatives: fes-ho

So in this sentence, ho sap is the normal order.

Sap is the third person singular present indicative of saber.

The present tense goes like this:

• jo sé
• tu saps
• ell / ella / vostè sap
• nosaltres sabem
• vosaltres sabeu
• ells / elles / vostès saben

So l'oncle ho sap means the uncle knows it.

Because Catalan distinguishes between two kinds of to know:

• saber = to know a fact, piece of information, or how to do something
• conèixer = to know a person, place, or be familiar with something

Here the sentence is about knowing some information, so saber is the correct verb:

• No sé si l'oncle ho sap = I don’t know if uncle knows it

If you were talking about knowing a person, you would use conèixer instead.

Because Catalan, like English, often leaves out repeated information when it is obvious from context.

So:

• però espero que no = but I hope not

The full idea is something like:

• però espero que no ho sàpiga = but I hope he doesn’t know it

Catalan does not need to repeat everything if the meaning is already clear.

Because after esperar que when expressing a wish, hope, or something uncertain, Catalan normally uses the subjunctive.

So:

• l'oncle ho sap = indicative, a statement of fact
• espero que no ho sàpiga = subjunctive, something hoped for / not certain

That is why the shortened version espero que no is useful: it avoids repeating the whole clause, but the implied full clause would still normally have the subjunctive.

They do different jobs:

• The first No negates sé:
  No sé = I don’t know
• The second no belongs to the clause after espero que:
  espero que no = I hope not

So the sentence is not a double negative in the English sense. Each no negates a different part of the sentence.