Tier 4 to Spouse Visa

Could a person on a Tier 4 visa marry a British Citizen?
Do they need to transfer to Spouse visa prior or after marriage?

Well they can't transfer to a spouse visa before marrying as they won't qualify as a spouse. After the wedding they can apply to switch to a FLR(M) visa, assuming they meet the financial condition of £18,600 earnings per annum (this can be joint). Also required, A1 level English and the payment of the new NHS Surcharge of £500 in addition to the visa fee of £649.
Bear in mind that the Registrar is now legally bound to notify the Home Office of the registration to marry and they could extend the notification period to marry from 28 days to 70 days and ask both to attend an interview before the wedding can take place.