For some explaining of why 100 miles isn’t far in Texas, at least.

Texas = 696,241 km² (269,000 square miles)
UK = 243,610 km² (94,060 square miles)

The oldest towns in the two countries?

The oldest community within the United States is Acoma Pueblo, 70 miles west of Albuquerque. The Acoma people have been living there for 900 years and still live there.

The oldest community still with inhabitants in the UK is Amesbury, Wiltshire. It’s been continually occupied since 8820 BC. That’s almost 3 millennia.

Oldest building still in use for generally the same purpose?

St. Peter of the Wall, in Essex, was built between 660 and 662 and is still being used as a church. It is 1350 years old.

The oldest church in the United States is San Miguel Mission in Santa Fe, New Mexico. It dates from 1710, replacing a 1626 chapel that burned. It is still a Catholic church.

Old public places?
The Old Bell Hotel in Malmesbury, Wiltshire was built as an inn in 1220. It will be celebrating its 800th anniversary soon. It’s still a hotel in the Cotswolds.

The oldest bar still in use as a bar in the US is Jean Lafitte’s Blacksmith Shop (and bar) from 1722. However, it was at least nominally a shop, rather than a bar. It will 300 years old soon.

Oldest shops?
The oldest shop in the UK is in Chester, Cheshire. It was built around 1274 as a shop and home. It’s still a shop. That’s 740 years.

The oldest shop in the US is in Moultonborough, New Hampshire. It was built in 1781 as a store and is still operating as a store. That’s 243 years.

Oldest gardens?
The oldest public garden in the US is from 1820.

The oldest public garden in the UK was first enclosed in 1415 and first had seats added in 1612. That makes it 405 years older than the US garden.

Oldest bed in use in the UK? 406 years.

The oldest bed found in the world is 77,000 years old. That makes the UK bed a baby, though people are still sleeping in it.