A lot will depend on signl strength and quality in your area. A simple two-way splitter would be cheap and effective if the signal from your aerial is adequate. I assume that you mean that the otput from your aerial is routed to your lounge, rather than actually being in it. In this case, if you have a mast head amplifier or equivalent fitted the power supply would need to be applied on the aerial side of the splitter as these do not pass DC.
I would be inclined to purchase a splitter and insert it at the TV end of the coax to test the impact on reception. If this is satisfactory, choose a point in the coax run to insert it that is both convenient and minimizes the length of cable needed to reach your set in the bedroom.
Amplified splitters are also available but may not be necessary in your case. I assume you are already using the freeview service. Signal strength will, of course, improve post switch-over, if that has not already occurred in your locality.