http://www.rusemblon.org

I checked this link and it said that for UK citizens applying for a tourist visa, you need to provide:
- Current passport which should be valid for a minimum of six months after visa's expiration date and has at least 2 blank pages.
- Visa application form filled in and printed out from this website only.
- One recent passport size photo glued to the indicated space on the application form.
- Original Tourist Voucher from a Russian (British) travel agency stamped and signed by an authorized person. [I'm confused about this - is this the voucher I have to ask the hotel to provide? - NorthOrSouth]
- Tourists on board large cruise Ships should have the right to stay in the Russian Federation without a visa during 72 hours. They have the right to leave the ship and come back on board the ship only as part of an organized tourist group as per the list. In case a tourist wants to leave the ship alone, he needs a visa.
- Bank statements for the last three months showing self-employed or unemployed applicants have minimum �100 per day for duration of visit. [Really - I have to provide this? - NorthOrSouth]

Then under 'General Requirements', it says '- A tourist contract - tourist voucher and tourist acceptance confirmation of a Russian tourist agency, which should have its reference number.' So I get the tourist voucher from the hotel or somewhere else? And what then is the tourist acceptance voucher.

I'm really confused. Can someone help me with what the steps for what I should do for an independent visit with flights booked independently (Austrian Airlines if it matters) and award stay at the Marriott Royal Aurora.

When should I apply? I'm going in early July? We have time to go to the visa centre to submit and collect our passports and documentation but we have a couple of short breaks planned elsewhere so I'm trying to plan when we should apply for the Russian tourist visas.

Thanks in advance for any help. I want to get this right.

You will get both documents (typically on the same page) if you request visa support from the hotel.

Yes. Your hotel or travel agency will give you these documents. I'm traveling to Russia with a US citizen resident in the UK who applied for her visa under the new process in London.
Our documents came on two separate pages - one "Accommodation Voucher" in English and one "Подтверждение" (Podtverzhdenie) or confirmation in Russian. The voucher contains her name, the travel agency name and location, hotel name and location, arrival date, and departure date. It is signed and stamped by the travel agency representative. The подтверждение states single entry, her citizenship, the dates of her entry and exit, her name, birthday, and passport number, the purpose of her visit, and places she will go on her visit. It is stamped and signed by the travel agency representative, along with stating the travel agency's registration and reference numbers with the Ministry of Foreign Affairs. They probably won't be too picky on whether it's one or two documents, as long as they contain all that information. They accepted e-mailed scans of the documents both at the London processing centre and at the Embassy in Washington, DC.

PM me if you want to see what my documents look like. (I can black out the personal information.)

Only if you're unemployed or self-employed.
Once you've got your hotel booked, call the hotel. If you give them your details, which you may have done anyway in the booking process, they should be able to get you a tourist voucher and confirmation by mail, fax, or e-mail. If you live in London or Edinburgh, it should only take a few days to walk it through in person, but I'd leave a week to be without your passport unless you pay for rush processing. I would also recommend applying at least two weeks ahead of your departure date if possible so that you can check your visa and get it corrected if there are any errors.
Though this doesn't affect you, the Russian Embassy in Washington refuses to issue visas that start more than 90 days from the date of submission of the application, at least for tourist visas, but I don't know if the UK processing agency imposes that requirement.