Where can a U.S. citizen get a Russian visa in Europe?

The US Embassy website on Russian Entry Visas clearly states that:

U.S. citizens who apply for Russian visas in third countries where they do not have permission to stay more than 90 days may face considerable delays in visa processing.

Unfortunately, you will have major delays if you apply for a Russian Entry visa at a Russian Embassy in Europe. Especially if you are in transit (less than 90 days).

You will not be allowed entry into Russia without the visa. The visa is easiest obtained at the Russian Embassy in Washington D.C. or (related consulates) within the USA. I go to the Houston consulate to get mine. They simply forward my paperwork to the Russian Embassy in Washington DC. The paperwork is not very easy I must say.

According to ILS Houston's FAQ website, a "rush" visa costs the same and takes 3 business days.

"You may submit documents at the Russian Visa for Consulate Institution consideration on a possibility of issuing a rush visa (up to 3 business days).

Please note, that the decision to issue a visa on expedite basis is taken by the Ambassador or Consul General of the Russian Federation."

This means the major guy in charge (the Ambassador! a foreign diplomat!) reviews your case. You need a good case to convince them you need to get to Russia fast. It is at the Russian Federation's discretion to allow you to "rush" an application. They will want to know your purpose and intent.

Also, keep in mind, you will have to surrender your passport in order to obtain the visa. So a minimum of 3 business days they will hold your passport in Washington D.C. to put a visa in it.

Where can a U.S. citizen get a Russian tourist (I think) visa while in Europe?

And to answer the original question...you can apply for a Russian entry visa at any Russian Embassy in the world. You can also view the official Russian Embassy website's list of them here.

In the UK, they don't have ILS, they have VFS Global instead, for all visa processing. You can review the documents required in the UK for USA nationalities here. If I were you, I would contact VFS Global, as soon as possible, to see if you comply with the requirements.

The current requirements for US Citizens are:

1. Current passport which should be valid for a minimum of six months after visa's expiration date and has at least 2 blank pages which do not need to be consecutive.
2. Visa application form filled in and printed out from this website only. Kindly ensure that the dates of your entry and exit, as entered in the application form, match with or fall within the period specified in the supporting documents.
3. One recent passport size photo glued to the indicated space on the application form.
4. A 'Tourist voucher' and a 'Tourist confirmation' document stamped and signed by authorized person. They can be obtained through your hotel or an approved travel agent who deals with trips to Russia, and should be valid for the entire duration of your trip. The stamp and all details should be clearly readable.
5. Tourists on board large cruise Ships have the right to stay in the Russian Federation without a visa for up to 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 Tourists who wish to leave the ship on their own or in a group of less than 5 tourists must have a visa.
6. Self-employed, company directors, working from home or unemployed applicants need to provide bank statements for the last three months which have a current balance of a minimum of £100 per day for the duration of the visit. Please note that if you wish to provide online banking printouts please make sure they will be certified and stamped by your bank. Students, house-wives and retired people are not required to provide bank statements.
7. Applicants going for auto tourism should enclose the Registration Certificate of the vehicle they intend to travel by and covering letter mentioning the details of the journey, itinerary and names of the accompanying travellers. Notice: Tourist Visa is issued according to the dates specified in the voucher and confirmation and can be for a maximum period of 30 days. Kindly ensure that the dates of your entry and exit, as entered in the application form, match with or fall within the period specified in the supporting documents.

Here are the current VFS Global processing times for Russian Visas:

RVAC London: Express service: 1 up to 3 working day; Standard service: 5 up to 20 working days;

RVAC Manchester: Express service: 1 up to 3 working day; (From the day received by the Russian Embassy in London) Standard service: 5 up to 20 working days;

RVAC Edinburgh: Express service: 1 working day; Standard service: 5 working days;

Also keep in mind, these rules change, adjust and (d)evolve over time. This is valid advice as of 7-30-2018. Next year, the rules could be different.