So let's say we have a guy in 1750 in the area we now call Kentucky. He wants to write a letter and send it to his long lost cousin in London.
How long would the journey take? What kind of mail systems were set up in that time. What dangers could be expected for mail carriers along the way and was there any sort of security that your letter would or would not arrive? How often did mail travel across the ocean?
Edit: Thanks to /u/breecher for showing the difference between a letter and a telegram. I specifically meant a handwritten letter.
I just want to point out that there seems to be some confusion between the thread title and question asked. A letter and a telegram are two very different forms of media.
The telegram, as understood as being the electrical telegram developed in the first half of the 19th century (and not the optical telegram which was developed in the late 18th century but only rarely used for personal correspondance) was a revolutionary method of transmitting messages almost instantaneously through wires, while the postal services transmitting letters was only ever as fast as the current means of transportation allowed them to be.
OK no telegrams in 1750 they didn't come in until the 1840s.
You would write a letter and ask someone travelling to the coast to mail it for you. There was a government postal service but it was mainly for government business and was not always available.
Anyway, you would have to get someone to take the letter to a sea port and ask a ship's captain to take it to London. They would probably get a messenger boy to deliver the letter once they dropped anchor in London.
Incidentally when government postal service was available, it was up to the recipient to pay the postage not the sender. And in the 1750s it could be quite expensive even for a letter.
The penny post and regular mail service was not established until the mid 19th century and rural free delivery some time in the late 19th century.
As to how long it would take. Figure 20 miles a day over land, on foot or on horseback. Then 4 to 6 weeks at sea, if the winds were favorable (they tend to be seasonal, you would find more ships going from east to west in summer, the opposite in winter). The last bit from the docks in London to his cousin's house, an hour or two.
If you were a government official or had 'pull' you could use the government mail service. Otherwise, you could ask but they did not have to take your letters. Some people were forbidden to use the post if the post master had a grudge against them. Ben Franklin had a problem getting his letters and papers delivered, when a rival printer got appointed postmaster and forbid him to use the post. When Franklin became postmaster he made sure the service was open to everyone.
As to private couriers there was no guarantee but sea captains delivered letters as a regular thing and were considered trustworthy. You would have to go by the reputation of the captain. And if they handed a letter to a messenger, the messenger would be counting on a nice tip when he delivered it.
There was always the chance a ship would be sunk in a storm but this did not happen too often. In the case of an important message you could send 2 copies by 2 different routes.