Heathrow Terminals 3, 4, 5 and overnight lodging

Arriving in Terminal 3, you will change to a different Heathrow Express train for Terminal 4. Listen carefully or ask someone to make sure you get on the correct train. At Terminal 4, you have a covered walkway to your hotel(same walkway as to Hilton). You can reverse the process to get to Terminal 5 the next morning-it's the same Heathrow Express train as Terminals 2,3. However, you will need to be at Terminal 5 at 7:30 AM and you may find it worth the taxi fare to go directly to Terminal 5 from your hotel-they can tell you how much it would be(I think I paid 20 pounds for a taxi from the Hilton in Terminal 4 to Terminal 3 earlier this month) and arrange the taxi for you the night before. If you don't take the taxi, you need to allow an additional 20-30 minutes in case there's a wait for the train.

You might consider the Yotel at Terminal 4 in the airport. Rooms are small, but have the basics and it saves you time by staying right at the airport. We stayed there several years ago when we had a late flight into London and an early flight out the next morning. Would definitely use them again for the convenience.

Beginner's LHR question here. I've changed planes there twice. The last time we had a similar connection issue. We just stayed in London for a few extra days, but if we had wanted to stay at LHR, there are hotels there???

Another vote for the Yotel at T4. Yes, it's minimal but you won't really have time or energy to haul yourselves further afield and back again the next morning. It will offer you a shower & bed, plus alarm clock and early tea or coffee if needed. Have all you need for overnight in a single bag; no room for unpacking more than that.

From Terminals 2 or 3. (also central area and bus station)
to: Terminal 4. Take the TFL Rail service. Alternatively use the 555 (19 minutes) bus from outside arrivals. <<

In the morning

From Terminal 4.
to: Terminal 5. Take the TFL Rail service to the central area and then transfer onto the Heathrow Express to terminal 5. Alternatively use the 482 (12 minutes), 490 (15 minutes) or the 724 (12 minutes) bus from outside arrivals.<<

I found the bus easier; the Yotel staff will tell you where to catch it. All these transfer services are free.
Check-in for your morning flight (yes, the buses do run early enough), and then look for breakfast airside after going through security.

Although the Yotel would make the departure morning simpler, when I investigated Heathrow options last year, the Yotel was more than twice as expensive as the Premier Inn, and the OP is going to need two rooms. And of course the PI rooms are considerably larger.

The Yotel cannot sleep 4 in one room.

There is a Sofitel basically in Terminal 5. (Okay, just to the side but you don't have to go outside to reach it.) A little more expensive than the other options given but it's a five minute walk to check in/security the next morning. I've stayed there a few times and it's very nice.

Apologies: I failed to notice this is for a family of 4. I agree that in that case the Yotel would not be the best (great for a solo traveller, though). At the other end of the scale, the Sofitel at T5 is indeed ideally situated; I found it too luxurious (and expensive) for my taste, given how few hours I was going to be there.
You might find this page useful: https://www.londontoolkit.com/lhr/heathrow_family_rooms.htm
(the search facility mentions a "quad room" at the Sofitel), and this for Sofitel prices https://www.londontoolkit.com/lhr/sofitel_lhr.htm

My husband's default for flights out of Heathrow is always the Hilton at Terminal 4(yes we have hotel points we can use.) It really is helpful to already be at Heathrow when you have a flight the next morning.

Whenever I have a connecting flight at Heathrow, I stay at the Sofitel Hotel at Terminal 5. It is outside security, but an easy stroll inside from the terminal.

"if we had wanted to stay at LHR, there are hotels there???"

Yes, as you can see from this thread, there are hotels actually connected to various terminals. There are also hotels near the airport, on a bus route (the Hotel Hoppa); these are obviously less convenient.

Although hotels at terminals that are not the one you are flying from also have a degree of inconvenience.