Would I be allowed to get on a plane from the UK and fly to America with tea?

You'll certainly be allowed to get on a plane in the UK with the tea, but you'll have to declare it to Customs and may have it examined by an agricultural specialist, who will decide whether it is allowed in.

Here's US Customs and Border Protection on importing tea:

There are no restrictions on the importation of coffee, tea or spices for personal use, although they may be subject to Animal and Plant Health Inspection Service (APHIS) and Food and Drug Administration (FDA) inspections if a Customs and Border Protection (CBP) Officer suspects contamination or insect infestation of the product. For more information from FDA call 1-888-723-3366. Contact FDA.

Please note that tea or spices containing fruit or vegetable leaves or seeds, including citrus leaves or seeds, are typically prohibited.

On the plane, you'll be given a Customs declaration form to fill out. On that form, it will ask if you are bringing in any plants, seeds, food, etc... Always declare any food or agricultural items you are bringing into the country. You can be fined for failure to declare. Someone from CBP may ask to see the tea, and there's a chance you won't be allowed to bring it in, but you shouldn't be otherwise penalized as long as you've declared it.

You'll likely have more success with commercially packaged tea rather than a random unmarked bag of loose leaf.

I have brought commercially packaged tea into the USA on numerous occasions without any incident. Didn't want to leave a half-empty box to be thrown out by the hotel maids.

On the other hand, I can not imagine why the OP thinks that tea is not a big deal in the United States. There are at least three high-end tea shops in Berkeley, California alone. (Site, Site, no relationship other than infrequent customer.) Most brands of tea I have seen marketed in the UK are available here; if the OP has a particular preference I can look.

A distinctive brand of tea will be useful if you are kidnapped by Fascist agents, though.

I always tell my relatives to bring their own teabags when they visit. There have been no issues whatever about this; customs officials think this perfectly normal for visitors from the UK, and may even look askance at those who do not.

Tea is HUGE here. You can get Earl Grey with lavender accents and snobby French teas that even the French can't pronounce and things done with "'erbs" and fruit that don't even have any connection with the tea plant and white tea and green tea and it's all served iced, with or without ice, or hot in a glass, probably with lemon on the side. Ask for milk in it, and people will look at you like you just drop-kicked a kitten, but if you're lucky they might be able to find a sachet of powdered non-dairy creamer that's normally used for coffee.

"A mug, white, with one" has no meaning here in the US, because the incredible range of things they call tea precludes any such simplicity. Even being able to find plain black tea (which is what most Brits expect when we say "tea") will be a challenge, as most places don't even stock the stuff.

While you can get brands of tea that claim to be the same, at specialist places like World Market and such, odds are that you won't find a reliable source of tea that tastes as you are accustomed until you've been here a while (few years maybe).

Mostly, they're as much the "same" as cola canned in the UK, US, and middle east are the "same", which is to say, they have the same logo on the box but there it ends.

So bring a good stash of your own comfort drinks.

Also, bring a travel kettle that can work on 120v. Not only do most hotels not have a kettle in each room, but it's hard to find an electric kettle that actually works for any reasonable amount of money.

And don't use tap water to make the tea. I know, "don't drink the water" is so hackneyed travel advice, it's become a trope. But either buy bottled water, or get a very good water filter (which, thankfully, they DO sell here). Mains water is typically laced with chlorine and fluoride by law, to the point that you step out of the shower smelling olike you just stepped out of a swimming pool. So you will meet many Americans with thyroid issues (I never even knew thyroids existed, beyond a vague "heard the name", before I came to the US), and none of them have any clue why. Most assume it's genetic.

TL;DR: Don't drink the water. Bring tea bags. Bring a 120v kettle. And be prepared for the fact that the toilet paper is both a different shape, and not up to BSI Kitemark standards. Frankly, you might as well wipe your arse on a teabag. But that's another rant, for another time.

And don't get me started on the plumbing.

This answer was too long for a comment but it just a combination of Zach's and Andrew's answer.

You shouldn't have a problem bringing in "packages" of tea. Try to stick with a sealed, store bought, container. An unopened box, with that plastic film around it for example. It will come down to the discretion of customs people rather your tea looks suspect or not. Tea bags in a sealed box, wrapped in plastic. Probably not going to be an issue. Loose leaves in a ziplock baggie, rolling around in your carry on..... might be a problem. Technically your only real concern is going to be some over-zealous customs personal or maybe a TSA agent that is really just clueless. In all honestly, no one will probably even look at it.

Different states have different rules. Usually though this is only going to be an issue if you bring an entire suit-case full of tea. Some states require that anyone carrying any kind of fruit or agricultural product stop and declare it. Again I have never seen this used to stop a person from carrying a "normal" amount of tea. I have however seen this used to stop a person from carrying single units of fruit. So you may want to be aware of that too.

Let me stop and be clear at this point. In both cases listed above, I don't really see how you could get in trouble for carrying tea. I don't even get how someone would even know, under normal circumstances. That said, weird things can happen and an agent who is on a power trip (happens a lot more then it should) may just give you hard time over it. Avoid metal containers though. Your problem, if you have one will probably come from; "Hey that guy has a bomb in his luggage, no wait it's just tea, well find some way to hassle him over it."

Next point however is that you should consider just getting your tea here. If you're going to a rural area you may need to stop in town on your way, but most cities will have what ever kind of tea you want at normally sane prices. If you know someone where you are going ask them to look around for you ahead of time. Big box stores may not have what you want, but there are usually tea shops around that certainly would.

I've never had any problems bringing back packaged loose leaf tea from Sweden or Austria. I've made the trip three times, and not once has customs asked to examine my tea. (They confiscated a single orange once though.) As one poster said, you will need to fill out a form on the plane declaring your items, but as long as you aren't bringing a boatload of tea, you should be fine. (My most recent trip had me bringing home approx. 2.5 pounds of Swedish tea.)

As an aside that doesn't answer the question... Yes, most readily available, loose leaf, American tea is horribly, horribly overpriced. I don't know about tea from the UK, but tea from Sweden is amazingly good compared to the two major tea stores I know here in Boston.

As a concrete datapoint, I have flown at least three times into the US where a full suitcase contained (branded, packaged) Argentinean yerba mate, say some 25kg. I declared it, the bag was opened but not the packages, and nothing happened.

My wife has brought tea from China many times. The inspector doesn't care one bit when she declares "tea". We have frequently gotten a suitcase inspection (common for travelers from China, even more common for ethnic Chinese travelers from China) and they have never cared about the tea, either.