travel Heathrow to Cotswolds

Take the coach to Oxford and spend the night there is my suggestion.

Take the coach to Oxford and spend the night there is my suggestion.

It would usually be mine too, but the OP is concerned about having traveled from Australia and being jetlagged, and her plane lands (if on time) at 6:40 pm. I'd prefer to head for a hotel room in that situation, instead of another leg of travel.

I went to booking.com to search for hotels around Heathrow. You can apply several filters - including your price range, do they include an airport shuttle, highly rated, and more - to winnow down the options to something manageable. I don't have experience with any of those hotels so I won't recommend any of them. You didn't say what you consider "budget" accommodations for an overnight near Heathrow, but as noted, you can narrow down your choices by price when searching booking.com. (I've never booked anything through booking.com; I book directly with hotels, after using booking.com and other sites to find them.)

There are a couple Premier Inns by the airport. It is a good budget chain. Premier Inn London Heathrow Airport T2 & T3 (Bath Road) hotel, Premier Inn London Heathrow Airport Terminal 4 hotel, and Premier Inn London Heathrow Airport Terminal 5 hotel. There’s also ibis budget London Heathrow Terminal 5, ibis budget London Heathrow Central, and Hotel ibis London Heathrow Airport. Ibis is another budget chain.

You don't say where in Oz you will be starting from, but if it is Melbourne and you are flying on Royal Brunai (a very comfortable inexpensive airline) you will be landing at Heathrow Terminal 4. If you land at T4 there is an above ground pedestrian tunnel from the terminal to 4 hotels at various price points. First there is a combination Crowne Plaza / Holiday Inn Express (very nice blue light bridges, I enjoyed staying there very much) and then a Hilton and then a Premier Inn.

If you land at a different terminal you can use the free trains between terminals.

Arriving that late, another vote for booking accommodation in or around Heathrow. As already mentioned there are several Premier Inns which have consistently high standards of accommodation. They also offer a good night's sleep guarantee!

I have stayed often at the Premier Inn T4. It typically has extraordinarily low rates. Definitely worth checking out if you don't want to spend more than necessary that night. Do be sure to choose the right type of reservation; the last time I checked there were three different types of cancellation policy.

What airline are you flying and in which terminal do you arrive?

I read that it is best to go straight to the Cotswolds from the airport

Did you happen to read that in a Rick Steves guidebook? It's not the "best" idea, it's his opinion. Mostly because that's where his tour starts.

Unless you opt for one of the few hotels at one of LHR’s terminals (likely to be more expensive), I would jump on the Elizabeth line to Paddington and look for a hotel close to there at your budget. Then when you wake up the next day (which might be any time given jet lag), you are right at the place to catch a train to the Cotswolds.

The T4 Premier Inn is usually a great deal cheaper than a comparable room in downtown London.

The Crowne Plaza at T4 on a Friday night last month was only £97. The Holiday Inn Express was about £10 less.