TLDR; Unless you are lucky, you will go through immigration twice. Therefore, you definitely need a single entry visa to cover one of your arrivals, but as you've a relatively short connection in HCMC, I'd consider paying the extra $25 stamping fee and getting a multiple entry one as a visa on arrival at Hanoi. The first agency I found on Google suggests a total price of $73 for this, I suspect you can find it a little cheaper.
First - there's a chance, albeit slim, that Angkor Air will be willing to through-check your bags. They're owned by Vietnam Airlines, which would make me optimistic, but you'll be dealing with someone whose English is limited at best, is probably rather busy, and who may never have done this before. Still worth asking, it'll save you a bit of faff in any case. But we can't rely on it.
As mentioned in Malvolio's answer, Brits don't need a visa to enter Vietnam - once. Unless you're spending 30 days in Siem Reap, you'll need a single entry visa for one of your visits. Now the Vietnamese imply in the link that it could be for either visit, and I've got anecdotes that suggest this is true. However, Timetic - the visa system used by the airlines - suggests otherwise:
Holders of British passports endorsed "British Citizen" entering Viet Nam without a visa must have a gap of at least 30 days from their last departure date from Viet Nam.
(you can do a search on Timetic with United, or through IATA )
I believe Timetic to be inaccurate, but it's probably safer in terms of being let onto the aircraft if you got any single entry visa for your second trip.
I'm sure it's covered in more detail elsewhere, but you've got two options for getting a visa. You could get a visa pre-approval letter (aka Visa On Arrival, aka VOA), or you could get a visa in advance from the Vietnamese Embassy in London.
Visas from the embassy look to be more expensive than a multiple-entry VOA visa, so we can rule that option out entirely. They don't publish fees online, but this travel company suggests it's about £60 ($80 right now) for a single-entry visa. For a VOA you pay a small fee to the visa company, and then currently $25/$50 at the airport for a 1-month single/multiple entry visa.
However, the problem with a pre-approval letter to be used on your second trip would be that it could easily add an hour to your transit time at HCMC as you'll need to queue up and wait for the visa, and then queue up again and wait to be let through the border. With a three hour gap between flights - which only gives you two hours before checkin closes - this would make me a little nervous. Although you'd probably be fine, perhaps even 95% of the time, I'd rather not spend the last bit of my holiday clockwatching.
So. You can get a single entry visa when you arrive for the first time at Hanoi, and risk Cambodia Angkor Air not understanding the actual visa requirements and denying you boarding in Siem Reap. You can get a single entry visa when you arrive for the second time, at HCMC, and risk missing your flight back to London (but take the chance of saving $25 by not needing to pay the airport bit of the fee if you do get through-checked). Or you can get a multiple entry visa the first time you arrive at Hanoi, but definitely pay more money than you need to.