The actor playing the Prince asked exactly the same question, when we were mounting the Romeo production in 2004. Director Tim Carroll had a simple response: 'act'! Indeed, if actors rely on their accent alone to convey a character, something has gone badly wrong. That's one of the irritating stage legacies of Received Pronunciation: I've often been told about actor 'laziness' - that all one has to do to convey a posh character is to sound posh, and the accent will do all the work. And conversely, of course, that all an actor has to do to play a lower-class character is to sound rustic. There's so much more to it than that.
The question betrays a misunderstanding of what OP is. OP is a phonology, not a phonetics. In other words, it represents the sound system of an earlier period in the history of English. Just as Modern English phonology has an indefinite number of phonetic realizations, so does Early Modern English phonology. In other words, there are several accents in OP. When we performed Romeo at the Globe, we had a Scottish-tinged Juliet, a Cockney-tinegd Nurse, a Northern Ireland-tinged Peter, and so on. But everyone reflected ths OP system in the way they spoke - for example, saying musician as 'mu-si-see-an', or pronouncing /r/ after vowels.
So of course there would have been differences between different parts of the country, and between the way the court and city people spoke and the way people spoke in the countryside. Indeed, Shakespeare says as much, in As You Like It, when disguised Orlando notices the way Rosalind speaks: 'Your accent is something finer than you could purchase in so removed a dwelling' (3.2.329). But what was that court accent? Was it 'like today'?
It was nothing like RP. RP evolved as an upper-class accent towards the end of the 18th century. In Elizabethan times, you could have a strongly regional accent and still reach the highest levels in the kingdom. We don't know exactly how Elizabeth spoke, but Francis Drake and Walter Raleigh were from Devon, and the judge Thomas Malet observed of the latter: 'he spoke broad Devonshire to his dying day'. When James brought his court down from Scotland, suddenly Scottish accents were everywhere. Francis Bacon describes James's speech as 'swift and cursory, and in the full dialect of his country'. And we know from observations such as occur in John Day's satirical 'The Isle of Gulls' (1606) that people would copy the discourse of the court. That play may even have been presented with both Scottish and Southern accents, judging by the observation of Sir Edward Hoby, in a 1606 letter, that 'all men's parts were acted of two diverse nations', and that - evidently King James didn't like it - some of the actors ended up in prison for their pains.
At the same time, we know from a famous quotation in George Puttenham's Art of English Poesie (1589) that poets were recommended to use 'good southern' - 'the usual speech of the court' or of the surrounding area to about 60 miles, and to avoid those from the north and west who used 'strange accents or ill-shapen sounds'. If there was no RP, what might this have been? We get a clue from Holofernes, who is thought to be based on a real-life spelling reformer, such as Richard Mulcaster. He knows how to read and write, and insists on pronouncing every letter. It's a natural process, still encountered today, when people pronounce the /t/ in often because, they say, 'it's there in the spelling'. So, for example, a court accent would almost certainly have pronounced the /h/ at the beginning of a stressed syllable, because it's there in the spelling. And that assumption immediately allows us to explore some interesting dramatic contrasts in OP. When A Midsummer Night's Dream was performed in OP at Kansas University a few years ago, the court characters and lovers kept their /h/'s and the mechanicals dropped them (unless trying too hard, in their play scene), as did the fairies. That allows Puck to emphasise the /h/'s when he is copying Demetrius and Lysander - a posh 'manhood', for example.
What all this points to is the existence of 'educated regional' accents, much as we have today. 'Modified RP', as some would say - or, even more recently, 'modified General English'.