I will be arriving late at Heathrow T2 from FRA (LH)and will be leaving the next morning from T3 to Singapore (SQ). I will spend the night in one of the LHR hotels. Do I have to pick up the luggage in LHR only to check it in again the following day or will through-checked baggage solve this problem?

(I'm travelling on a combined ticket)

Another question. What is the probable connecting time from T2 to T3 for a First Class passenger?

Thanks in advance