I think you will be OK, you can presumably get your wife to confirm to an agent airside that you are all present and correct (it should show up in Ready To Fly if she gets airside before you do), and I would hope the landside agent would be helpful to you. The worst that could happen is that you could be pinged for the charges for one suitcase, £65, and even if that did happen then I imagine you could reclaim it later. You only save £5 by doing it online.

But obviously it would help if you can get at least Sapphire status somehow (though it does show in your profile?) to get 2 bags free, or get the travel issued from a travel agency on one ticket, which normally allows through baggage. Also your wife/child may be allowed by a Border Force agent to transit without a visa, depending on a vast range of factors. Or upgrade yourself to CE in return for all the heavy lifting. The car seat / stroller should not be charged for with a child on the booking (even if HBO!).