Not much to add to the good advice already here. UK and EU passports have a pretty easy time in Southeast Asia, but yes, I would get the Vietnamese via ahead of time. The Vietnamese Embassy in Bangkok is easily accessible, on Wireless Road, close to a couple of Skytrain stations over on Ploenchit Road.
Thailand does require a visa, it’s just that you don’t have to get it ahead of time; they have what is called “visa on arrival.” And I believe you can get a longer visa for Thailand if you get it ahead of time than if you just show up. There has been talk of eliminating visa fees for the next couple of months or so in a bid to entice tourists back into Thailand after last year’s debacle with the airport closures. That would save about 1000 baht (US$28), although truth be told if saving just 28 bucks is what it takes to persuade people to visit Thailand, I’m not sure they’d have enough money to live on, cheap as it is here.
The visa situation in Laos is good right now. And tomorrow (Thursday), there will even be a new train service between Thailand and Laos, the first one ever, from our Nong Khai in the Northeast to Thanaleng Station outside of Vientiane, the Lao capital, a short hop. I would be careful about traveling overland between Vientiane and Luang Prabang, as there have been some bus attacks. We’ve only ever flown into Luang Prabang. The Lao Embassy in Bangkok is rather difficult to get to, so check ahead of time about visas, but I believe visa on arrival is the norm right now. In fact, there are lots of agencies in Nong Khai that will handle the visa for a fee.
The Cambodian Embassy used to be in a great spot in central Bangkok but recently moved out by the Lao Embassy. Should not be a problem with visa on arrival for Cambodia, though.
Malaysia and Singapore are both too easy; don’t even think about getting a visa ahead of time. Malaysia MAY be a problem if there’s any evidence of travel to Israel, but I think they’ve chilled out; they used not to let Israeli citizens in, but I believe that’s changed.
EDIT: You mentioned a month in Thailand. I’m on an annual non-immigrant visa myself, so I’m not always up on the tourist visas, but I believe the standard now is 30 days when flying in to Thailand but only 15 days if coming in overland. I would call the local embassy just to ask. Extending a tourist visa in-country costs 1900 baht ($53), so it’s worth knowing ahead of time exactly how long the visa will be good for. But I’m sure it’s at least 30 days.