Check out https://www.between9and5.com/en/c1591803/day_rooms_London/
A day room accommodates bags and showers in one swipe, you get to pick the location (to some extent), the kids can take a nap and its pretty convenient.
It may be a little more expensive than a locker + public-shower option but probably not by much and the convenience should be significantly better.
I’m not sure there’s a good solution here, it’ll depend if you want to maximize time or minimize money. Heathrow is off in the wrong direction, leaving the bags there is going to burn between 90 minutes and a few hours collecting them so I doubt it’s worth it.
Honestly, I’d consider a cheap hotel/B&B in the Euston or Paddington area. You could even reach out to a few of them in advance and explain you just need somewhere to leave luggage and have a shower, they may be able to help, it can’t be that unusual. That also gives you the option of somewhere to rest if you need it.
The main problem is that all the left luggage places in the stations (at least the rail station part) are run by the same company and so charge largely the same prices — £10 per item/day.
Going to Euston and leaving your bags there would be the obvious good plan. As mentioned in the question and the comments you could use the lounge there or the showers at Waverley (as mentioned, £5 per person).
However Heathrow to Euston isn’t that easy an journey, if you’re taking public transport the TFL Journey Planner is suggesting the Heathrow Express to Paddington.
But, if you’re going via Paddington you may as well leave your bags there (same price) and there are showers there, again same price.
Or you could go via Tube to St Pancras/Kings Cross left luggage is the same and there are showers at Kings Cross.
Note that all those luggage services close at 11pm, so you’d have to be back there in enough time.
If the luggage price is too steep your other option is Victoria Coach Station (as mentioned in the answer linked in the comments). There luggage is £4 to £6 per bag. I can’t find showers in either the Rail or Coach station at Victoria, but you’d still have the option of Waverley or Euston. You could get the National Express Coach direct to Victoria. Luggage closes at 10:45pm but you’d need to be there beforehand to make it to Euston in time.
Overall, it depends how much you want to spend, how much time you want to have free and how much luggage you have.
If money is no object then Heathrow Express to Paddington, leave bags there (optionally shower), get the tube to go see some things, get back to Paddington before the left luggage closes and take a taxi to Euston. But, if I were doing that I’d be very tempted to check out hotels near there, there are a lot within walking distance that are really pretty cheap.
Cheapest option I can see (that’s not just carrying your bags all day) would be National Express to Victoria.
And hopefully everything above gives you some insight as to the choices in-between.
I’m trying to think if there’s some ‘out of the box’ solution, but it’s escaping me. There may be some travel package you could book that would take care of the luggage — that’d be a question for a travel agent somewhere.
Partial answer: I would store your luggage in central London, if you store it at the airport you’ll spend a large proportion of your day going back and forth to Heathrow.
There is a left luggage storage company at Euston, though it’s expensive at £10 per item.
Credit:stackoverflow.com‘