Originally Posted by Scots_Al

On 1, Transport for London has a good journey planner you can play around with at:

https://tfl.gov.uk/plan-a-journey/

I put in LHR Terminal 5 to Vauxhall at 1630 on a random weekday later this month. The fastest result was 46 minutes: Heathrow Express; Bakerloo Line to Oxford Circus; Victoria Line to Vauxhall.

I then edited preferences to remove National Rail results, and got a 59 minute journey by underground: Piccadilly Line to South Kensington; Circle/District Line to Victoria; Victoria Line to Vauxhall.

To be honest, neither sounds very appealing at that time of day with kids and baggage in tow. Might be worth looking into how much a pre-booked car might be.