Added warning: read the whole answer before you read the first lines which discuss road transport.
Rome2Rio says the road part of this journey will take an hour and a half. Going by taxi or private car will halve the time, and double the cost.
Remember that the time estimates do not include time to stand in line to clear immigration and customs at Luton, nor time to get to the bus stand at Luton, nor time to stand in line to get from the underground station to your terminal at LHR, nor time to stand in line to check in, nor time to stand in line to pass security. And it's possible your airline wants you to check in at LHR earlier than you'd arrive even if everything went swimmingly.
Your tickets are probably independent, as I doubt any airline would sell you these two flights on one ticket because of the significant chance you'll miss the LHR connection.
You can see where I'm going with this. This is a very tight connection. Because there's a significant chance you'll miss the LHR flight (and therefore have to buy a new ticket to your destination and maintain yourself until the new flight), I think this is a bad bet, and I wouldn't attempt it.