Not possible with heatmap. Here's a solution using surf.
% Inputs
data = [ 1 2 3; 4 5 6];
rowWidths = [10, 20]; %mm, one for each row of data; applied bottom to top
colWidths = [5 15 20]; %mm, one for each col of data; applied left to right
% Produce figure
fig = figure();
ax = axes(fig);
% Plot image
[xg,yg] = meshgrid([0,cumsum(colWidths)], [0,cumsum(rowWidths)]);
sh = surf(xg, yg, zeros(size(xg)), padarray(data,[1,1], 0, 'post'));
% Set colormap (this one is same as heatmap)
n = 255;
cmap = [linspace(.9,0,n)', linspace(.9447,.447,n)', linspace(.9741,.741,n)'];
colormap(ax,cmap)
cb = colorbar(ax);
axis tight
axis equal
view(2)
xlabel('mm')
ylabel('mm)')
To add text labels similar to heatmap,
% find rectangle centers
xCnt = xg(1:end-1,1:end-1) + diff(xg(1,:))/2;
yCnt = yg(1:end-1,1:end-1) + diff(yg(:,1))/2;
th = text(xCnt(:), yCnt(:), compose('%g',data(:)), ...
'HorizontalAlignment','center',...
'VerticalAlignment','middle', ...
'FontSize', 10, 'Color', 'w');
If you want the axis size to reflect the axes scales (ie, if you want the axis size to be 40x30mm), you'll need to set the axis size before plotting. It will look something like,
% Compute axis and figure size
axisSize = [sum(rowWidths), sum(colWidths)]/10; %cm, [width,height]
margins = 1; %cm margin size
figSize = (axisSize + margins*[2,2]); % cm
% Produce figure
fig = figure('Units','centimeters');
fig.Position(3:4) = figSize; % note: there is a min fig size
movegui(fig)
ax = axes('Units','centimeters',...
'Position',[margins,margins,axisSize]);
