Recommendations for family traveling with 3.5 & 5 year olds for about 3 days in London/Dover?

In the UK car seats must normally be used for children until they’re 12 years old or 135cm tall, whichever comes first.

https://www.gov.uk/child-car-seats-the-rules/using-a-child-car-seat-or-booster-seat

Perfectly safe to take small children on public transport, though crowded busy tube stations are best avoided at peak times.

An ambitious holiday with children of that age!

With such short notice, you need to book your accommodation first rather than planning your activities first. The best value rooms will already have been sold. Public transport is good so you can be located anywhere in the city centre. Use buses as well as tubes as they are easier to board with children. Take HS1 train from St Pancras to Dover as it is quicker.

If you have to carry your children, you would be better off with bags with wheels rather than back packs. There are not usually any left baggage facilities other than at the occasional rail station and these will not be near the attractions that you are likely to be visiting.

Thank you two, I appreciate the feedback :) The carseat link was great--I was able to read the rules and traveling without them on tour buses won't be an issue so I want to book these excursions next! Actually, I think that's all I have left;) because...

we just booked the hotel on Bath Road near the airport for all three nights so we can store our luggage while touring for the day. And, I had forgotten way back in March, I had ordered two Heys hard-shell children's suitcases specifically designed for ride-on during travel, so that might not end up being too bad after all anyway! Promo here: https://www.youtube.com/watch?v=kBBw5Nwxkj0

Now just looking for the best bus tours=) Thanks again!

Was just going to say, the Bath Road near Heathrow is a LONG way out of tourist London... but emma beat me to it. Totally not where I'd stay. The air around the airport is pretty bad also, and the noise.

On a lighter note... food.

I'm not a veggie myself and don't even know anyone who is, but it is something that's catered for in London, albeit on a small scale. Almost everywhere will have some basic veggie options, and they are good at marking menu's with a green 'V' to signify a veggie dish.

Pret-a-Manger have a reasonable range of salads, sandwiches, etc. and their food is always good. Strangely, the one and only thing I've ever had from Pret that I didn't enjoy was a veggie falafel wrap, I love falafel, but this one wasn't crispy and I didn't enjoy it... no texture.

Kebab shops will tend to have a veggie falafel offering on the menu, and they are usually fresh and cheap.

The king of veggie food in the UK has to be Indian. They have loads and loads of super tasty veggie dishes and they can all be as mild or spicy as you like, just ask for mild, medium or hot when ordering. I like meat in my food, but I can happily have a veggie Indian meal and not miss the meat at all. The chickpea (chana) and lentil (dhal) dishes can be very 'meaty' even though there is none.

You need to cancel your accommodations. Too far away from London and you'll be wasting two hours each day in travel time. Simply not a good idea to "maximize your adventure." Immediately start searching on Airbnb and VRBO websites for accommodations that are in travel zones 1&2. I'd also check the Falcon Hotel, the Coach House http://www.coachhouse.chslondon.com/, for your dates.
Secondly head to a bookstore and peruse the R S London guide. Will be helpful to you in understanding London and how to maxmize your 3 days.

Once again, I really appreciate the feedback! I was able to cancel our reservation for the first leg of the visit based on your comments and am now beginning the search for a more suitable place given our situation. Any suggestions are greatly appreciated:)

I have had good luck with booking.com as well. You can take whatever is available and they do usually have a flexible cancellation policy, so if you find something else within a few hours should be able to choose the best one... but, it is really short timing, so I would jump on whatever looks halfway reasonable, or you may end up way out again.