Setting in Austen's novel has two large components of time and place to it, though with variations to place. One wants to say the setting is critical to this novel, but is it? Setting certainly is vague in terms of descriptive qualifiers:
Mr. Bingley had not been of age two years, when he was tempted by an accidental recommendation to look at Netherfield House. He did look at it, and into it for half-an-hour—was pleased with the situation and the principal rooms, satisfied with what the owner said in its praise, and took it immediately.
Let's examine setting's importance to the central theme of pride and prejudice and to the secondary themes of marriage and socio-economic status.
The theme of pride and prejudice can be acted out in any time period and any locale. In fact it has been in the Indian Bollywood film Bride and Prejudice. As another example, we might even have a group of sisters living with family in New York City who have various jobs and who get an interesting collection of neighbors on their floor in their high-rise apartment building. Thus the major theme is not locked in importance into the time and place of Jane Austen's era. What about secondary themes?
These two are harder to see as successful outside of Austen's setting. Marriage then was essentially a requirement as Charlotte demonstrates through her explanations to Elizabeth and through her marriage to Collins.
"I ask only a comfortable home; and considering Mr. Collins's character, connection, and situation in life, I am convinced that my chance of happiness with him is as fair as most people can boast on entering the marriage state."
Women were confined to a sparse range of roles unless they were heiresses (with property and wealth free of entailment) or widows, who had autonomy, authority, wealth and power, like Lady de Bourgh. So for Jane Austen's commentary on marriage, the setting is very important and this theme is locked into the setting: it can't be played out as well in any contemporary Westernized setting (although there may conceivably be some non-Westernized setting in this contemporary historic period where Austen's marriage theme might work out).
Socio-economic status is so different in today's world (thanks mainly to educational opportunities and women's rights)--even the caste system in India is being obliterated--that I can't see this theme being successful outside of Austen's setting. Perhaps a setting incorporating globalization (Elizabeth the King's daughter in an African tribe ...) might allow for the theme of wealth and power to be expressed in the same way. Yet generally, it seems this secondary theme is also locked in importance to Austen's setting.
In addition, it is conceivable that all three of these themes could successfully move back in time to the Western Renaissance or even the Middle Ages, as socio-economic models of these eras might be overlaid with Austen's model with some ease and success.