Transit visa

You require a single entry Chinese tourist (L) visa because of your first itinerary. Taking a Transmongolian train requires a visa obtained in advance. Your second stop is eligible for 24 or 144 hour visa free transit in direct relation to your length of stay in Beijing. If your entire transit time will be less than 24 hours, you will be eligible for 24 hour visa free transit and if your stay will be longer foir just a minute than 24 hours, you will enjoy 144 hour visa free transit.

By the way as a UK citizen, you are eligible for a longer term, multiple entry visa, probably with 2 years validity, so you are encouraged to have this detail in mind and to ask for this long term visa declaring that you intend to travel China more often in the near future.