Deletion of /h/

I was once told by my BrE teacher that when we have a word ending with a consonant followed by another word starting with h, the h is deleted, meaning that the pronunciation is different.

For instance, must have should be pronounced as /'mʌstæv/ and not /mʌst hæv/. The same for look at her should be pronounced as /'lʊkətə/ and not /lʊk ət hɜː/.

Does this stand for BrE speakers?

According to a pronouncing dictionary that I've owned for 40+ years, it's perfectly-correct British English to drop the H, except after a pause or when particularly emphasized, in the word the pronouns 'he', 'him', 'her', and in the word 'have' when unstressed.

In many cases the initial "h" will not be silent after a consonant in common US speech. This very morning I told my waiter that, "I will have green ham and eggs". I pronounced both "h" clearly. The gentleman showed no raised eyebrow at my manner, and indeed my ham and eggs arrived safe and sound, and tasty, and, of course, quite green. Note, however, that in some parts of the British Empire some particular subject might well have said, "I'll 'ave me green 'am an' eggs." But I am no expert in who says what over there!