Denver to London to Penzance

The basics - and more - of UK train travel are well explained here: https://www.seat61.com/

For UK rail journeys, the travel planner here is IMO the place to begin: https://www.nationalrail.co.uk/

Trains from London to Penzance leave from Paddington Station. They are operated by the Great Western Railway: https://www.gwr.com/ and I recommend you buy tickets directly from GWR vs. a third party seller like trainline.

I've taken the GWR in the opposite direction: the direct train from Penzance to London. It's listed as a 5 hour journey but it seemed longer. I'm glad I treated myself to First Class. Had my own single seat/table and appreciated the extra space and comfort. Even in first class you'd be sleeping sitting up if you do fall asleep. But no worries if you do fall asleep on the direct train: Penzance is literally the end of the line.
(If you can hang around London till late the night of your arrival, you could take the sleeper version of that journey to Penzance: https://www.seat61.com/sleeper-to-cornwall.htm and https://www.gwr.com/travelling-with-us/night-riviera-sleeper. It doesn't sound attractive to me as an option for your day of arrival off the DEN-LHR flight, but I'm mentioning it anyway.)

I'll leave the ticketing advice to our experts on here.

If you're flying BA from Denver you'll land around 11:45 AM. The simplest rail option would be the Heathrow Express (HEX) to Paddington, then the GWR train to Penzance. https://www.heathrowexpress.com/ (There's some hate usually expressed on here for the HEX whenever it's mentioned, but especially if you can get an advance £5.50 ticket online it's a fine way to get to Paddington after your flight.)

I'd probably take that train if I wanted to get to Penzance ASAP after landing at Heathrow. Not sure if I would take it the afternoon that I land, but that's a personal choice based on my level of tiredness from the long flight. I'm traveling to London in May, and going to Edinburgh right away. I've booked a room at a hotel at King's Cross for the night of my arrival. I'll relax for the afternoon and evening of my arrival, and then catch a morning train to Edinburgh after a good night's rest.

People would differ on whether to have a gentle day on the day of arrival in London and travel the next day, bearing in mind that it is a 5 1/2 hour journey.
I would argue for the onward journey that day, but only you know your body. Many others here would say take it gently, which would also be a buffer against a severe flight delay.
Main line trains don't call at Heathrow so you have to catch either a Heathrow Express or Elizabeth Line train into London Paddington, then a direct train to Penzance back out of Paddington. (Elizabeth Line is half the price of HEX and only marginally slower). Trains to Penzance leave hourly at 4 minutes past the hour until 1804 (6.04 pm) from Paddington.
Or catch the Rail Link coach to Reading then the same train from Reading to Penzance (it's first stop). Personally I don't see the point of that over going into Paddington, as it saves no time, may even be slower.
You can buy a ticket on the day (separate tickets Heathrow to Paddington, and Paddington to Penzance)- trains don't sell out. But you will pay an awful lot more and risk having to stand for a large part of the journey, so advance reservation is strongly recommended. On the day it is £109, book ahead and you may get a ticket for £43, more likely around £60.
You don't say what time you arrive, or which terminal, but I would suggest catching the direct (no change) National Express Coach.
It takes longer but no changes, and Penzance is the last stop.
Departures from Heathrow Bus Station (at Terminals 2 and 3) are at 0925, 1130 and 1500, arriving at 1840, 1955 and 0005
respectively. Book an amendable ticket, not the cheapest price fixed ticket.

EDIT (after @ Suz's thoughts) - on the Night Riviera sleeper (if booked well in advance), leaves Paddington just before midnight, arrives Penzance just before 8am- you can get a cabin for 2 for an all inclusive price of £237. But I too would really question that after a TA flight.

Do not want to purchase tickets in advance. Do trains fill up or I we OK with purchasing tickets on the day of arrival?

Train tickets don’t sell out (with the exception of Euro star). But you will pay far more on the day. A Two together Card or Senior Rail Card (if eligible) may save you money.

PS- Thanks to @ Suz for the extra info. For Tomorrow it says the BA218 flight arrives at 0830 at Terminal 5, and the UA 027 at 0940 at Terminal 2, so maybe the BA times differ by season- I defer to Suz on that.
So it would be useful to know if it was BA or United. If it's BA the two trains go direct from T5, whereas for the coach you need to take the free shuttle train (the same train as to Paddington) the few minutes to T2/3- the 1st stop.
I'm reading what those who do the TA flights say about getting out of Heathrow afterwards, and would be bit concerned about making a 1130 coach off an 0830 flight arrival, and certainly off the 0940, but others could advise better on that.

I've been really impartial in what I wrote above re- HEX/Elizabeth Line, trying to lower the tone in an oft repeated debate. It's your money, your choice. I think the word 'disdain' rather than 'hate' is better in Brits views towards HEX.

... catch the Rail Link coach to Reading then the same train from Reading to Penzance (it's first stop). Personally I don't see the point of that over going into Paddington, as it saves no time, may even be slower.

It is generally somewhat quicker from T5 but more of a wash from T2/3. Was somewhat a better option when there were special fares that were flexible on the day but more like the Advance price level but these disappeared some time. Today the best way value wise is to book the coach separately if choosing this way but it is rather better for a return than a single trip.

I gave up using the Rail Air LInk from Reading on my trips to Heathrow prior to Covid
Its normally quicker and cheaper to go via London
Some journey planners suggest Elizabeth Line train to Heathrow to Hayes and Harlington and then change at Slough and or Reading for Penzance

Thanks so much for your input. I think I will do what I originally thought of doing which is to visit London first for 4 days and then take the train to Penzance from Paddington. I will book the train ahead of time. Seems to be a lot less stressful. Last year we were caught up in all the airline cancellations and delays so I am leery about trying to catch the train when we get there because who knows when we will actually arrive. Last year British Airways and United both cancelled flights on us. Hopefully, this year will be better.

That sounds a much more relaxed plan.